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Modeling of paint flow rate flux for circular paint sprays by using experimental paint thickness distribution
Mechanics Research Communications, Vol. 26, No. 5, pp. 609-617, 1999 Copyright © 1999 Elsevier Science Ltd Printed in the USA. All rights reserved 0093-6413/99/S--see front matter
Pergamon
PII S0093-6413(99)00069-5
MODELING
OF PAINT FLOW RATE FLUX
FOR CIRCULAR EXPERIMENTAL
PAINT SPRAYS BY USING PAINT THICKNESS
DISTRIBUTION
Tuna BALKAN and M.A. Sahir ARIKAN Mechanical Engineering Department, Middle East Technical University ln6nti Bulvan, 06531 Ankara, Turkey
(Received26 March 1999; acceptedfor print 21 July 1999) Introduction
In spray painting, paint is transferred to the surface by making use of an atomizing medium, which is air spray. Air streamlines radiate from the nozzle of the spray gun, and a paint flux contained within a cone is obtained. Paint, generally mixed with thinner, is stored in a pressurized paint container and delivered to the spray gun by making use of the pressure. Another supply to the spray gun is the atomizing air, which is used to form fast air streams at the paint nozzle tip. These streams create low pressure and turbulence, and consequently atomize the paint. Because of non-uniform flux distribution within the cone, a non-uniform coating thickness occurs on the painted surface. Besides the technical specifications of the spray gun, air and paint nozzles, and paint needle; basic settings like paint tank pressure, spray air pressure and gun needle-valve position affect paint cone angle and paint flow rate flux. Paint flow rate flux, together with the geometry of the surface, spray gun path, and painting velocity, should be used for the analysis and computer simulation of the spray painting process. The spray gun is attached to an industrial robot so that the desired spray path and painting veloci.ty can be obtained easily. In this study, paint flow rate flux is determined by. using experimental paint thickness distributions obtained by using different spray gun settings and painting parameters. Various theoretical and experimental studies have been made on prediction or determination of paint thickness distribution. A tabular technique for determination of the spray pattern is developed by Goodman and Hoppeusteadt [1]. The thickness of a test pattern is measured at a grid of points, and a least-squares technique is used to fit an expression to the measured points. An automatic trajectory planning system for spray painting robots is developed by Sub et at. [2]. In the study, a simple coating analysis is made and an elliptical paint thickness distribution is found.
609
610
T. B A L K A N a n d M.A. S A H I R A R I K A N
In the study by Persoons and Van Brussel [3], a simulation tool to predict the thickness of coating layer on highly curved surfaces is presented. The model proposed in their study makes use of the diffusion equation to define the concentration of the paint particles in the non-homogeneous mixture of paint particles and atomizing air dispersion. When predicted results of the model for paint thickness distribution are compared to the experimental results, the model is found to be less accurate because of not considering the flow-out of the paint on the surface due to the internal pressure of the spray cone atomizing air. An experimental paint thickness distribution resulting from a single painting stroke applied on a flat surface is also given in their study. Although not stated, the distribution is practically parabolic.
Paint Thickness Distribution and Paint Flow Rate Flux The first step for the computer simulation of the spray painting process is to model the paint flow rate flux, which determines the paint thickness distribution on the surface to be painted. On the other hand, when the paint thickness distribution resulting from a single painting stroke applied on a fiat surface is known, then the paint flow rate flux which causes the distribution can be found, and can be used for simulation of painting of more complex surfaces. There are many factors that affect the characteristics of the paint flow rate flux through the spray gun. These factors can be grouped as, technical specifications of the spray gun, air and paint nozzles, and paint needle; settings like paint tank pressure, spray air pressure and gun needle-valve position; and properties like amount of thinner in the paint, paint temperature and viscosity., and workpiece temperature. In many studies made to model the paint flow rate flux by considering the above factors, theoretical results obtained do not agree well with the experimental results. There are a number of reasons for this disagreement. Spraying process is a complex process, and all of the above factors cannot be considered easily. Devices like pressure gages and pressure regulators used in painting equipment are of average quality, i.e. imprecise and most probably inaccurate. Control valves used on spray guns for adjustment of paint, spray air and fiat spray air flow rates and for setting the direction of the fiat spray air flow are simple valves, too. Since they do not have any scales, their settings are not repeatable. These factors make it difficult to quantify., and precisely and accurately set and measure all of the parameters that affect the process. Although the settings of the pressure and flow rate control valves are kept fixed, properties like amount of thinner in the paint, paint temperature and viscosity, and workpiece temperature affect the paint flow rate flux and paint thickness distribution. Due to the above reasons, when an effective model to predict the paint flow rate flux cannot be developed, painting process cannot be simulated and the paint thickness distribution on a complex surface cannot be predicted correctly. Thus, determination of the paint flow rate flux by using experimental paint thickness distribution resulting from a single painting stroke applied on a fiat surface becomes a feasible approach. As long as the spray gun settings and painting parameters are not changed, the paint flow rate flux does not change and can be used for the computer simulation of painting of more complex pans. On the other hand, effect of spray distance on paint thickness distribution can easily be investigated and included in the expression of the paint flow rate flux. In the previous studies on paint thickness; elliptical [2], parabolic or Gaussian [3] distributions have been either proposed or found experimentally. In this study, in order to make a general solution, and to be able to model various paint thickness distributions, the Beta distribution is utilized. As seen in Figure 1, paint thickness distribution resulting from a single painting stroke applied on a flat surface can be expressed as follows by making use of the Beta distribution.
F L O W IN P A I N T S P R A Y S
y
611
yii: L
w ~
~
"~ Spray area
Figure 1. Single Painting Stroke and Resulting Paint Thickness Distribution
T(x) = T~× (1 - 4x2 )~-1 _ w/2 < x _
(l)
W~
In the above equation, T is the paint thickness, T~.~ is the maximum thickness, w is the width of the painted zone, and also the diameter of the spray area for circular spray areas. Tm~, w and can be found by using experimental data and the least squares curve fitting technique. Circular spray areas are obtained by closing the flat spray air valve of the spray gun. When this valve is opened, elliptical spray areas are formed. In this study, only circular spray areas are considered. Distance x is measured in the width direction starting from the location of the maximum thickness, i.e. from the middle of the painted band. For the same w and Tm,~, various distributions can be modeled by using different 13's. When 13=1.5 the distribution is elliptical and when 13=2 the distribution is parabolic, With larger 13's the Beta distribution can also be used to model the Gaussian distribution. Figure 2 shows plots of T(xJ3) for various f3's for Tm~~ 100 p.m and w=80 ram. =
1,l.Tm 100 T(x, 1.5) T(x,2) T(x, 3) T(x, 5)
50
T(x, 10) T(x, 20)
,,,
,
oL 40 w 2
"-_ 20
0 x
20
40 w 2
Figure 2. Effect of 13 on Beta Distribution
612
T. B A L K A N a n d M . A . S A H I R A R I K A N
Paint thickness distribution on a flat surface resulting from a single painting stroke of the spray gun can be found by making use of the Cartesian coordinate system attached to the center O of the circular spray area, which is on the centerline of the spray gun. During the painting process, this coordinate system moves with the spray gun with velocity v. As shown in Figure 1, y-axis is in the direction of the gun motion, and x-axis is perpendicular to the y-axis to form a righthanded coordinate system. Paint thickness distribution along x-axis can be found by considering a point P, which is at a distance x from point O. As seen in Figure 3, at time t=-h P is just started to be painted, and at t=h, since P is no longer within the spray area, painting ends. In order to make use of symmetry, the time interval is taken between -h and h.
X
X
X
w
t = t~
y=-yo r = w/2
t =
t = -ta
y=O r=x
Y= Yo r = w/2
Figure 3. Painting of Point P
Within tile period -tl to h, since y coordinate of P continuously changes, its radial distance r measured from O also changes. At any time t, y can be expressed as follows. y = -vt
(2)
At t=-tl and at t=tl the radial distance of P is equal to the radius of the circular spray area w/2, and at t=0 it becomes equal to x. Time tl can be calculated as follows by using the spray gun veloci.ty v, and starting value of y , Yo. tl = Y.q_o V
where,
(3)
613
FLOW IN PAINT SPRAYS
yo = ~/(w 2 / 4 ) - x 2 .
(4)
During the painting process x does not change, and y can be calculated by using Equation 2. At any time, radial distance r of point P measured from O can be found by considering its x and y coordinates. r=~
= ~ x 2 + v2t 2
(5)
Resulting paint thickness at point O is found by integrating the unknown paint flow rate flux q(r) within the period -tl to h. T(x) = tlq(r)dt = 2ttq(r)dt tI
(6)
0
Time t can be written in terms of r by using Equation 5. 2_x
l
=
-
2
(7)
-
V
Then, dt=
r v r2~_x 2dr,
and, 2w. '2 r T(x) . -- . J ~. . . v × q r - _ x . q(r)dr
2 x r J ~ q ( r ) vwn%/r~_x ~
dr.
(8)
The above equation is a special type of the Volterra integral equation of first kind, and its general form and solution are as given below [4]. f(x) = i
~b(r) dr {p(x) - p(r)} ~
d?(r) = sin_~n d i p'(x)f(x____.~)d x n d r c { p ( r ) - p ( x ) } 1-a
(9)
(10)
In the above equations, f(x)=-v.T(x)/2, c=w/2, d~(r)=r.q(r), p(r)=-r 2, p(x)=-x 2, p'(x)=-2x, ~t=l/2, 1-or=l/2 and sin etn=l. The paint flow rate flux q(r) corresponding to an experimentally determined paint thickness distribution T(x) can be obtained as follows by using the thickness distribution given in Equation 1.
614
T. B A L K A N a n d M . A . S A H I R A R I K A N
q(r)=VT~
d ~
rw!2
x (1 - 4X2) ~'l dx w~
(11)
Finally, 'after solving the above equation numerically, the paint flow rate flux can be expressed again by using the Beta distribution with a different 13, which is denoted by 13'. qm,x and 13' can be found by using the numerical solution of q(r) and the least squares curve fitting technique. 4 r ~ 0 ' ~ , r ~w/2 q(r) = o~ . . . . tl ---7-J
(12)
W"
Sample Results
In order to model the paint flow rate flux, different painting strokes at different spray distances and painting velocities are made on a flat surface by using paint sprays with circular spray areas. These experiments are performed by using FANUC ArcMate Sr. industrial robot and BINKS 95A spray gun. Then, thickness measurements are made across the strokes after the paint dries out completely. For thickness measurements FISCHER Permascope M11D coating thickness gage is used. As a sample solution, paint flow rate flux corresponding to the data given in Table 1 is determined by using MathCAD ~ as shown in Figure 4. Measured paint thickness values at various points {x[mm], T(x) [~m]} are, (15, 55), (25, 52), (35, 51), (45, 54), (55, 49), (65, 45), (75,42), (85, 40), (95, 29), (105, 26), (115, 23), (125, 19) and (135, 13).
Table 1. Data for a Single Painting Stroke on a Flat Surface Paint tank pressure Spray air pressure Flat spray air pressure Spray distance Painting velocity, v
0.15 MPa 0.20 MPa Not used. 250 mm 100 mm/s
Discussion and Conclusion
Paint flow rate flux through the spray gun is one of the basic variables required to model the spray painting process and to predict the paint thickness distribution on the surface to be painted. This non-uniform flux distribution within the spray cone causes a non-uniform coating thickness distribution on the painted surface. Various studies have been made to find a theoretical model for the flow rate flux, but for most of the time results which do not agree well with the experimental results have been obtained because of theoretical and practical difficulties, as explained. In order to overcome these difficulties, a different, but a direct approach, which makes use of the experimental paint thickness distribution resulting from a single painting stroke
FLOW IN PAINT SPRAYS
615
INPUT DATA Input painting velocity v (mmls)
v = 100
Input number of thickness measurements
nrn
]5
13
~55 52
25 i
Input X (mm) and measured thickness T (pm) values
MAKE LEAST SQUARES FIT BY USING genfit FUNCTION
X
r 55 , Tmax(ttm) /
Make initial guess forTmax,[3 andw
g =
2
13
270
w (ram)
P : gent]IX, T. g, Fng) i 56
]
P ~ i 2.3
!
Tmax(p,m)
35
i51
45
54
J
55
49
(15
45
75
42
85
4(1
95
29
105
126
115
23
125
19
135
13
Tmax= Po
~
~Pl
w (mm)
w = P2
i
I 330.5 j
PLOT DATA POINTS AND CURVE FITTING [x, X (mm); t, T (p.m)] i
O..mn- 1
W
'
t(x)
W
V¢
x . . . . , - - + O . l .... 2 2 2 F
F
t(x) =Fng(x.P)0
F
, /
Ti
r
010
181.8
-90.9
0
90.9
1111.8
x,Xi
Figure 4. MathCAD ® Session for the Solution of the Paint Flow Rate Flux (Continued)
616
T. B A L K A N
and M.A. SAHIR ARIKAN
SOLVE FLOW RATE FLUX q(r) NUMERICALLY Input number of points for solution
I
qLr)
,i' 4.x2!~ x.!l - - - : L
v.Tmax d ~.r
dri
(\X2
n = 10 1
W2
dx r
2~ 0-5
[w .'2
i ~0.. n [O
2
2.n
ri
ro , i.ro
qi
q(ro ~ i.ro)
PLOT FLOW RATE FLUX q(r) [r (mm); q(p.m/s)]
20
'i
I
I I
10
~1 i
0
•
L.J
0
50
100
150
r
MAKE LEAST SQUARES FIT BY USING genfit FUNCTION Make initial guess
for qmax and13'
g:=
[30I
qmax (p.mls)
I3'
P = genfi(r, q, g, Fg)
P=I 24.311.8
13'qmax(~tm/s)
Figure 4. MathCAD ~ Session for the Solution of the Paint Flow Rate Flux (Continued)
F L O W 1N P A I N T S P R A Y S
PLOT DATA POINTS AND CURVE FITTING FOR q(r) [r, R (ram); q, qf (p.mls)]
i
0..n-2
20
-w W
W
R :--.---~0.1..2 2 2
qf(R)
Fg(R,P) o _
~
qf(R) qi OOg
,o 1 -181.8
i
-90.9
0 R,ri
90.9
181.8
Figure 4. MathCAD® Session for the Solution of the Paint Flow Rate Flux (Continued) applied on a fiat surface is used; and the flow rate flux causing the measured distribution is determined. If the spray gun settings and painting parameters are not changed, the flow rate flux does not change, and the obtained thickness distribution shows a very good agreement with the predicted thickness distribution when painting of a complex surface is simulated by using the determined paint flow rate flux. The developed model and method can also be used to determine the flow rate fluxes resulting from different painting equipment and/or painting system settings, and results can be saved in files for further usage. Calibration experiments can be made and settings of the system can be checked periodically by using the developed model and the method. On the other hand, as seen in Equation 12, expressing the paint flow rate flux by a compact and simple equation is very beneficial, since a large number of computations should be made during computer simulation of the painting process of a complex surface. References
1. Goodman E.D, and Hoppensteadt, L.T.W, 1991, "A Method for Accurate Simulation of Robotic Spray Application Using Empirical Parametrization", Proceedings of the 1991 1EEE International Conference on Robotics and Automation, pp. 1357-1368, April 1991, Sacramento-California, USA. 2. Suh, S-H., Woo, I-K. and Noh, S-K., 1991, "Development of an Automatic Trajectory Planning System (ATPS) for Spray Painting Robots," Proceedings of the 1991 IEEE International Conference on Robotics and Automation, pp. 1948-1955, April 1991, Sacramento-California, USA. 3. Persoons, W. and van Brussel, H., 1993, "CAD-based Robotic Coating of Highly Curved Surfaces", Proceedings of 24th International Symposium of Industrial Robots (1SIR), pp. 611618. 4. Chambers, LI.G., 1976, lntegral Equations : A Short Course, International Textbook Company Limited, IlK.
617
Pergamon
PII S0093-6413(99)00069-5
MODELING
OF PAINT FLOW RATE FLUX
FOR CIRCULAR EXPERIMENTAL
PAINT SPRAYS BY USING PAINT THICKNESS
DISTRIBUTION
Tuna BALKAN and M.A. Sahir ARIKAN Mechanical Engineering Department, Middle East Technical University ln6nti Bulvan, 06531 Ankara, Turkey
(Received26 March 1999; acceptedfor print 21 July 1999) Introduction
In spray painting, paint is transferred to the surface by making use of an atomizing medium, which is air spray. Air streamlines radiate from the nozzle of the spray gun, and a paint flux contained within a cone is obtained. Paint, generally mixed with thinner, is stored in a pressurized paint container and delivered to the spray gun by making use of the pressure. Another supply to the spray gun is the atomizing air, which is used to form fast air streams at the paint nozzle tip. These streams create low pressure and turbulence, and consequently atomize the paint. Because of non-uniform flux distribution within the cone, a non-uniform coating thickness occurs on the painted surface. Besides the technical specifications of the spray gun, air and paint nozzles, and paint needle; basic settings like paint tank pressure, spray air pressure and gun needle-valve position affect paint cone angle and paint flow rate flux. Paint flow rate flux, together with the geometry of the surface, spray gun path, and painting velocity, should be used for the analysis and computer simulation of the spray painting process. The spray gun is attached to an industrial robot so that the desired spray path and painting veloci.ty can be obtained easily. In this study, paint flow rate flux is determined by. using experimental paint thickness distributions obtained by using different spray gun settings and painting parameters. Various theoretical and experimental studies have been made on prediction or determination of paint thickness distribution. A tabular technique for determination of the spray pattern is developed by Goodman and Hoppeusteadt [1]. The thickness of a test pattern is measured at a grid of points, and a least-squares technique is used to fit an expression to the measured points. An automatic trajectory planning system for spray painting robots is developed by Sub et at. [2]. In the study, a simple coating analysis is made and an elliptical paint thickness distribution is found.
609
610
T. B A L K A N a n d M.A. S A H I R A R I K A N
In the study by Persoons and Van Brussel [3], a simulation tool to predict the thickness of coating layer on highly curved surfaces is presented. The model proposed in their study makes use of the diffusion equation to define the concentration of the paint particles in the non-homogeneous mixture of paint particles and atomizing air dispersion. When predicted results of the model for paint thickness distribution are compared to the experimental results, the model is found to be less accurate because of not considering the flow-out of the paint on the surface due to the internal pressure of the spray cone atomizing air. An experimental paint thickness distribution resulting from a single painting stroke applied on a flat surface is also given in their study. Although not stated, the distribution is practically parabolic.
Paint Thickness Distribution and Paint Flow Rate Flux The first step for the computer simulation of the spray painting process is to model the paint flow rate flux, which determines the paint thickness distribution on the surface to be painted. On the other hand, when the paint thickness distribution resulting from a single painting stroke applied on a fiat surface is known, then the paint flow rate flux which causes the distribution can be found, and can be used for simulation of painting of more complex surfaces. There are many factors that affect the characteristics of the paint flow rate flux through the spray gun. These factors can be grouped as, technical specifications of the spray gun, air and paint nozzles, and paint needle; settings like paint tank pressure, spray air pressure and gun needle-valve position; and properties like amount of thinner in the paint, paint temperature and viscosity., and workpiece temperature. In many studies made to model the paint flow rate flux by considering the above factors, theoretical results obtained do not agree well with the experimental results. There are a number of reasons for this disagreement. Spraying process is a complex process, and all of the above factors cannot be considered easily. Devices like pressure gages and pressure regulators used in painting equipment are of average quality, i.e. imprecise and most probably inaccurate. Control valves used on spray guns for adjustment of paint, spray air and fiat spray air flow rates and for setting the direction of the fiat spray air flow are simple valves, too. Since they do not have any scales, their settings are not repeatable. These factors make it difficult to quantify., and precisely and accurately set and measure all of the parameters that affect the process. Although the settings of the pressure and flow rate control valves are kept fixed, properties like amount of thinner in the paint, paint temperature and viscosity, and workpiece temperature affect the paint flow rate flux and paint thickness distribution. Due to the above reasons, when an effective model to predict the paint flow rate flux cannot be developed, painting process cannot be simulated and the paint thickness distribution on a complex surface cannot be predicted correctly. Thus, determination of the paint flow rate flux by using experimental paint thickness distribution resulting from a single painting stroke applied on a fiat surface becomes a feasible approach. As long as the spray gun settings and painting parameters are not changed, the paint flow rate flux does not change and can be used for the computer simulation of painting of more complex pans. On the other hand, effect of spray distance on paint thickness distribution can easily be investigated and included in the expression of the paint flow rate flux. In the previous studies on paint thickness; elliptical [2], parabolic or Gaussian [3] distributions have been either proposed or found experimentally. In this study, in order to make a general solution, and to be able to model various paint thickness distributions, the Beta distribution is utilized. As seen in Figure 1, paint thickness distribution resulting from a single painting stroke applied on a flat surface can be expressed as follows by making use of the Beta distribution.
F L O W IN P A I N T S P R A Y S
y
611
yii: L
w ~
~
"~ Spray area
Figure 1. Single Painting Stroke and Resulting Paint Thickness Distribution
T(x) = T~× (1 - 4x2 )~-1 _ w/2 < x _
(l)
W~
In the above equation, T is the paint thickness, T~.~ is the maximum thickness, w is the width of the painted zone, and also the diameter of the spray area for circular spray areas. Tm~, w and can be found by using experimental data and the least squares curve fitting technique. Circular spray areas are obtained by closing the flat spray air valve of the spray gun. When this valve is opened, elliptical spray areas are formed. In this study, only circular spray areas are considered. Distance x is measured in the width direction starting from the location of the maximum thickness, i.e. from the middle of the painted band. For the same w and Tm,~, various distributions can be modeled by using different 13's. When 13=1.5 the distribution is elliptical and when 13=2 the distribution is parabolic, With larger 13's the Beta distribution can also be used to model the Gaussian distribution. Figure 2 shows plots of T(xJ3) for various f3's for Tm~~ 100 p.m and w=80 ram. =
1,l.Tm 100 T(x, 1.5) T(x,2) T(x, 3) T(x, 5)
50
T(x, 10) T(x, 20)
,,,
,
oL 40 w 2
"-_ 20
0 x
20
40 w 2
Figure 2. Effect of 13 on Beta Distribution
612
T. B A L K A N a n d M . A . S A H I R A R I K A N
Paint thickness distribution on a flat surface resulting from a single painting stroke of the spray gun can be found by making use of the Cartesian coordinate system attached to the center O of the circular spray area, which is on the centerline of the spray gun. During the painting process, this coordinate system moves with the spray gun with velocity v. As shown in Figure 1, y-axis is in the direction of the gun motion, and x-axis is perpendicular to the y-axis to form a righthanded coordinate system. Paint thickness distribution along x-axis can be found by considering a point P, which is at a distance x from point O. As seen in Figure 3, at time t=-h P is just started to be painted, and at t=h, since P is no longer within the spray area, painting ends. In order to make use of symmetry, the time interval is taken between -h and h.
X
X
X
w
t = t~
y=-yo r = w/2
t =
t = -ta
y=O r=x
Y= Yo r = w/2
Figure 3. Painting of Point P
Within tile period -tl to h, since y coordinate of P continuously changes, its radial distance r measured from O also changes. At any time t, y can be expressed as follows. y = -vt
(2)
At t=-tl and at t=tl the radial distance of P is equal to the radius of the circular spray area w/2, and at t=0 it becomes equal to x. Time tl can be calculated as follows by using the spray gun veloci.ty v, and starting value of y , Yo. tl = Y.q_o V
where,
(3)
613
FLOW IN PAINT SPRAYS
yo = ~/(w 2 / 4 ) - x 2 .
(4)
During the painting process x does not change, and y can be calculated by using Equation 2. At any time, radial distance r of point P measured from O can be found by considering its x and y coordinates. r=~
= ~ x 2 + v2t 2
(5)
Resulting paint thickness at point O is found by integrating the unknown paint flow rate flux q(r) within the period -tl to h. T(x) = tlq(r)dt = 2ttq(r)dt tI
(6)
0
Time t can be written in terms of r by using Equation 5. 2_x
l
=
-
2
(7)
-
V
Then, dt=
r v r2~_x 2dr,
and, 2w. '2 r T(x) . -- . J ~. . . v × q r - _ x . q(r)dr
2 x r J ~ q ( r ) vwn%/r~_x ~
dr.
(8)
The above equation is a special type of the Volterra integral equation of first kind, and its general form and solution are as given below [4]. f(x) = i
~b(r) dr {p(x) - p(r)} ~
d?(r) = sin_~n d i p'(x)f(x____.~)d x n d r c { p ( r ) - p ( x ) } 1-a
(9)
(10)
In the above equations, f(x)=-v.T(x)/2, c=w/2, d~(r)=r.q(r), p(r)=-r 2, p(x)=-x 2, p'(x)=-2x, ~t=l/2, 1-or=l/2 and sin etn=l. The paint flow rate flux q(r) corresponding to an experimentally determined paint thickness distribution T(x) can be obtained as follows by using the thickness distribution given in Equation 1.
614
T. B A L K A N a n d M . A . S A H I R A R I K A N
q(r)=VT~
d ~
rw!2
x (1 - 4X2) ~'l dx w~
(11)
Finally, 'after solving the above equation numerically, the paint flow rate flux can be expressed again by using the Beta distribution with a different 13, which is denoted by 13'. qm,x and 13' can be found by using the numerical solution of q(r) and the least squares curve fitting technique. 4 r ~ 0 ' ~ , r ~w/2 q(r) = o~ . . . . tl ---7-J
(12)
W"
Sample Results
In order to model the paint flow rate flux, different painting strokes at different spray distances and painting velocities are made on a flat surface by using paint sprays with circular spray areas. These experiments are performed by using FANUC ArcMate Sr. industrial robot and BINKS 95A spray gun. Then, thickness measurements are made across the strokes after the paint dries out completely. For thickness measurements FISCHER Permascope M11D coating thickness gage is used. As a sample solution, paint flow rate flux corresponding to the data given in Table 1 is determined by using MathCAD ~ as shown in Figure 4. Measured paint thickness values at various points {x[mm], T(x) [~m]} are, (15, 55), (25, 52), (35, 51), (45, 54), (55, 49), (65, 45), (75,42), (85, 40), (95, 29), (105, 26), (115, 23), (125, 19) and (135, 13).
Table 1. Data for a Single Painting Stroke on a Flat Surface Paint tank pressure Spray air pressure Flat spray air pressure Spray distance Painting velocity, v
0.15 MPa 0.20 MPa Not used. 250 mm 100 mm/s
Discussion and Conclusion
Paint flow rate flux through the spray gun is one of the basic variables required to model the spray painting process and to predict the paint thickness distribution on the surface to be painted. This non-uniform flux distribution within the spray cone causes a non-uniform coating thickness distribution on the painted surface. Various studies have been made to find a theoretical model for the flow rate flux, but for most of the time results which do not agree well with the experimental results have been obtained because of theoretical and practical difficulties, as explained. In order to overcome these difficulties, a different, but a direct approach, which makes use of the experimental paint thickness distribution resulting from a single painting stroke
FLOW IN PAINT SPRAYS
615
INPUT DATA Input painting velocity v (mmls)
v = 100
Input number of thickness measurements
nrn
]5
13
~55 52
25 i
Input X (mm) and measured thickness T (pm) values
MAKE LEAST SQUARES FIT BY USING genfit FUNCTION
X
r 55 , Tmax(ttm) /
Make initial guess forTmax,[3 andw
g =
2
13
270
w (ram)
P : gent]IX, T. g, Fng) i 56
]
P ~ i 2.3
!
Tmax(p,m)
35
i51
45
54
J
55
49
(15
45
75
42
85
4(1
95
29
105
126
115
23
125
19
135
13
Tmax= Po
~
~Pl
w (mm)
w = P2
i
I 330.5 j
PLOT DATA POINTS AND CURVE FITTING [x, X (mm); t, T (p.m)] i
O..mn- 1
W
'
t(x)
W
V¢
x . . . . , - - + O . l .... 2 2 2 F
F
t(x) =Fng(x.P)0
F
, /
Ti
r
010
181.8
-90.9
0
90.9
1111.8
x,Xi
Figure 4. MathCAD ® Session for the Solution of the Paint Flow Rate Flux (Continued)
616
T. B A L K A N
and M.A. SAHIR ARIKAN
SOLVE FLOW RATE FLUX q(r) NUMERICALLY Input number of points for solution
I
qLr)
,i' 4.x2!~ x.!l - - - : L
v.Tmax d ~.r
dri
(\X2
n = 10 1
W2
dx r
2~ 0-5
[w .'2
i ~0.. n [O
2
2.n
ri
ro , i.ro
qi
q(ro ~ i.ro)
PLOT FLOW RATE FLUX q(r) [r (mm); q(p.m/s)]
20
'i
I
I I
10
~1 i
0
•
L.J
0
50
100
150
r
MAKE LEAST SQUARES FIT BY USING genfit FUNCTION Make initial guess
for qmax and13'
g:=
[30I
qmax (p.mls)
I3'
P = genfi(r, q, g, Fg)
P=I 24.311.8
13'qmax(~tm/s)
Figure 4. MathCAD ~ Session for the Solution of the Paint Flow Rate Flux (Continued)
F L O W 1N P A I N T S P R A Y S
PLOT DATA POINTS AND CURVE FITTING FOR q(r) [r, R (ram); q, qf (p.mls)]
i
0..n-2
20
-w W
W
R :--.---~0.1..2 2 2
qf(R)
Fg(R,P) o _
~
qf(R) qi OOg
,o 1 -181.8
i
-90.9
0 R,ri
90.9
181.8
Figure 4. MathCAD® Session for the Solution of the Paint Flow Rate Flux (Continued) applied on a fiat surface is used; and the flow rate flux causing the measured distribution is determined. If the spray gun settings and painting parameters are not changed, the flow rate flux does not change, and the obtained thickness distribution shows a very good agreement with the predicted thickness distribution when painting of a complex surface is simulated by using the determined paint flow rate flux. The developed model and method can also be used to determine the flow rate fluxes resulting from different painting equipment and/or painting system settings, and results can be saved in files for further usage. Calibration experiments can be made and settings of the system can be checked periodically by using the developed model and the method. On the other hand, as seen in Equation 12, expressing the paint flow rate flux by a compact and simple equation is very beneficial, since a large number of computations should be made during computer simulation of the painting process of a complex surface. References
1. Goodman E.D, and Hoppensteadt, L.T.W, 1991, "A Method for Accurate Simulation of Robotic Spray Application Using Empirical Parametrization", Proceedings of the 1991 1EEE International Conference on Robotics and Automation, pp. 1357-1368, April 1991, Sacramento-California, USA. 2. Suh, S-H., Woo, I-K. and Noh, S-K., 1991, "Development of an Automatic Trajectory Planning System (ATPS) for Spray Painting Robots," Proceedings of the 1991 IEEE International Conference on Robotics and Automation, pp. 1948-1955, April 1991, Sacramento-California, USA. 3. Persoons, W. and van Brussel, H., 1993, "CAD-based Robotic Coating of Highly Curved Surfaces", Proceedings of 24th International Symposium of Industrial Robots (1SIR), pp. 611618. 4. Chambers, LI.G., 1976, lntegral Equations : A Short Course, International Textbook Company Limited, IlK.
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