Granular Fertiliser Particle Dynamics On and Off a Spinner Spreader

Transcription

1 Granular Fertiliser Particle Dynamics On and Off a Spinner Spreader A. Aphale; N. Bolander; J. Park; L. Shaw; J. Svec; and C. Wassgren School of Mechanical Engineering, Purdue University, West Lafayette, IN , USA; of corresponding author: wassgren@purdue.edu Abstract A comprehensive experimental and analytical study was performed to investigate particle trajectories on and off a spinner spreader. Sixteen different granular fertilisers were used. Measurements of the significant material properties, on-spinner trajectories, and offspinner distance distributions were made. Analytical models for the on-spinner and offspinner trajectories were also presented and a comparison between the predictions and experimental results were made. The models reasonably predict, on average, the distance the real fertiliser materials travel.

2 Notation In the notation given below, the dimensions L, M, and T represent length, mass and time, respectively; primed quantities in the text are dimensionless A frontal projected area of a particle [L 2 ] C D drag coefficient for a spinner particle [-] g the magnitude of gravitational acceleration [L/T 2 ] g gravitational acceleration vector [L/T 2 ] F force applied to the rolling particle by the vane [ML/ T 2 ] H the height of the spinner plate above the ground [L] I moment of inertia of the particle [ML 2 ] K L aerodynamic resistance coefficient [1/L] the horizontal distance a particle travels from the rim of the spinner plate to when it contacts the ground [L] m n r r 0 mass of a fertilizer particle [M] particle radius [L] radial position from the center of the spinner disc [L] the radial distance from the spinner center to the location where a particle is dropped onto the spinner plate [L] R T spinner disc radius [L] the time required for a particle to move from the drop radius to the edge of the spinner disc [T] t v 0 time [T] the total velocity of a particle when it leaves the spinner plate [L/T]

3 v T particle terminal velocity in water [L/T] V the volume of a particle [L 3 ] x position vector of a particle from the edge of the spinner plate [L] α,β characteristic roots to the pure-sliding, on-spinner differential equation [-] µ friction coefficient between a particle and the spinner disc and a particle and a spinner vane [-] ρ mass density of a particle [M/L 3 ] ρ air mass density of air [M/L 3 ] ρ w mass density of water [M/L 3 ] τ the total time from when a particle leaves the spinner plate to when it contacts the ground [T] ω θ rotational speed of the spinner plate [1/T] angular position of the particle about its center of mass[-]

4 1. Introduction In recent years, greater emphasis has been placed on the proper handling and application of agricultural fertilisers in order to increase crop yield, reduce costs, and minimise environmental pollution. Granular fertilisers are the most common type of agricultural fertiliser since they are easily produced, transported, and applied. In order to be effective, however, granular fertilisers must be properly handled and distributed. One of the most common granular fertiliser application devices is the spinner spreader. The spinner spreader consists of a rotating disc with vanes bolted to the disc surface (Fig. 1). Fertiliser poured onto the spinner is thrown onto the ground after colliding with the rotating vanes. The chief advantages of the spinner spreader are that it has a large spread width, small size, simple and robust construction, and is inexpensive to produce. The performance of spinner spreaders has been widely investigated. Studies have typically focused on analytical models for particle trajectories on and off of the spinner. Several experimental studies have also been performed, but these often utilise idealised particles, typically steel ball bearings, or a limited number of granular fertilisers. An excellent summary of many of these studies, in addition to the significant material properties affecting fertiliser distribution from spinner spreaders, is given in Hofstee and Huisman (1990) and Hofstee (1992). Patterson and Reece (1962) investigated the motion of spherical particles on a spinner with a near-centre feed neglecting particle bounce off of the spinner vanes. They developed analytical models for on-spinner particle motion and found reasonable 1

5 agreement between their models and experimental measurements of the radial and total velocities of steel ball bearings leaving the spinner plate as well as the angle between where particles are dropped and where they leave the spinner plate. Inns and Reece (1962) developed a similar model for on-spinner particle motion, but considered an offcentre feed and particle bounce against the spinner vanes. Their model predicted the dynamics of steel ball bearings well but performed poorly for irregularly-shaped fertiliser particles due to the wide range of rebound angles. Cunningham (1963) developed models for various straight and curved vane configurations that were later found to be consistent with experimental measurements of the angle at which two different fertilisers leave the spinner plate (Cunningham & Chao, 1967). Hofstee (1994 & 1995) utilised an ultrasonic transducer to determine the velocity and direction of particles leaving a spinner spreader. He found that the friction coefficient between a particle and the spinner plate and vane is a significant variable; however, obtaining representative values for this variable for use in on-spinner models is difficult. Mennel and Reece (1963) developed an approximate model for off-spinner particle trajectories assuming turbulent flow. Although their model includes drag coefficients for both spherical and irregularly-shaped particles, they only compared their analytical results with experimental measurements using steel ball bearings. They report good agreement between the model and experiments but do not present the experimental data in their publication. Pitt et al. (1982) simplified the off-spinner equations of motion to generate an explicit analytical expression for the distance a particle travels after leaving the spinner plate. The error between the approximate solution and the models of Mennel and Reece (1963) is less than 7%. Using their explicit relation, they found that the 2

6 particle size distribution has little influence on the shape of the fertilizer spread pattern, but that the drop location of the particles does have a significant effect. An experiment using ammonium nitrate showed general agreement with the predicted spread pattern assuming a range of drop locations. The previously described models have been incorporated into comprehensive simulation studies by Griffis et al. (1983) and Olieslagers et al. (1996). These comprehensive models consider both on-spinner and off-spinner particle motion and ultimately predict the location at which particles land on the ground. Griffis et al. (1983) compared their predicted spread patterns with experiments using two different fertilisers and found large discrepancies. They attributed these to irregularly-shaped particles (the models assumed spheres) and particle-particle interactions. Olieslagers et al. (1996) also found a significant difference between their predicted spread pattern and an experimental distribution. They also attributed the discrepancy to particle-particle interactions, which they corrected for by adjusting the simulation input parameters to fit the experimental data. These previous investigations have greatly increased our understanding of how to model the dynamics of particles distributed by a spinner spreader. However, relatively few experiments, many of which utilise idealised fertiliser particles such as steel ball bearings, have been performed to validate the analytical models, especially using a wide variety of actual granular fertilisers. Our goals in this paper are to review the analyses for on- and off-spinner particle dynamics, and compare predictions from these models to experimental data for sixteen granular fertilisers. 3

7 2. Analyses Analyses of the granular fertiliser dynamics are presented in two sections: particle dynamics while on the spinner plate (on-spinner dynamics) and particle dynamics after the fertiliser has left the spinner plate (off-spinner dynamics). In both of these analyses, it is assumed that particles do not interact with one another Particle dynamics on the spinner plate A schematic of the geometry used to investigate the on-spinner particle dynamics is shown in Fig. 2. A flat spinner with radial vanes is used in this analysis since the experiments also utilize this geometry. A particle is assumed to fall onto the spinning plate against a vane, without bouncing, and with zero radial velocity at the drop radius, r 0. The analysis is presented for both pure-sliding and pure-rolling conditions Pure sliding Applying Newton s Second Law in the radial direction to the particle and considering centrifugal acceleration and sliding friction forces applied by the vane and the plate gives: (1) 2 mr = 2µ mrω+ mrω - µ mg where the overdots imply differentiation with respect to time t, m is the particle mass, r is the radial position of the particle, ω is the angular velocity of the spinner plate, µ is the particle/vane and particle/plate friction coefficient (the two friction coefficients are assumed to be equal) and g is the gravitational acceleration. Equation (1) is identical to 4

8 that originally derived by Patterson and Reece (1962). Note that the particle/plate friction force is included in the equation despite the fact that the ratio of the particle/vane friction force to the particle/plate friction force, 2 r ω / g, will typically be small over most of the time the particle is on the spinner. Assuming a constant angular velocity and applying the initial conditions r( t ) r t = 0 = 0 gives the solution: and ( ) = = 0 r0 r r0 1 1 µ g g exp 2 ( t) exp( t) µ = β αω α βω 2 R R β α + ω r 0 ω r0 r r0 αβ µ g = 1 exp 2 ( αωt) exp( βωt) ωr R β α ω r 0 (2) (3) where R is the spinner plate radius and the coefficients α and β are the characteristic roots for the pure sliding equation : 2 αβ, = µ ± µ + 1 (4) Pure rolling For the pure rolling case, the particle s rotational and translational speeds are not independent and are related by: θ n= r (5) where θ is the particle s rotational velocity and n is the particle radius. Newton s Second Law for the particle s translational and rotational motion are, respectively: (6) 2 mr = F + mrω µ mg I θ = nf (7) 5

9 where F is the force applied to the particle by the vane (the particle is assume to roll, and not slide, against the vane), µmg is the sliding friction force applied by the spinner plate, and I is the particle s moment of inertia. Combining Eqns (5) to (7) gives the following differential equation: 2 ω µ g r r = (8) 1+ I 2 1+ I 2 mn mn which is identical to that originally derived by Patterson and Reece (1962). Solving Eqn (8) subject to the initial conditions r( t = ) = r and ( ) 0 0 r t = 0 = 0 and assuming solid, spherical particles ( I mn 2 = 2 5) gives: r r0 µ g 5 µ g = 1 cosh ωt R R ω r0 7 ω r0 (9) r 5 µ g r0 5 = 1 sinh ωt 2 ωr 7 ω r0 R 7 (10) For both pure sliding and pure rolling, the time at which the particle leaves the spinner plate, T, occurs when r (t = T) = R and can be solved numerically for pure sliding using Eqn (2) or, for pure rolling, in closed form using Eqn (9). The total velocity of the particle when it leaves the spinner v 0, will include both radial and tangential velocity (=ωr) components: ( ) [ ω ] v0 = r t = T + R (11) The angular distance the particle travels from the drop point to where it leaves the spinner plate is simply ωt. 6

10 2.2. Particle motion off the spinner plate A schematic of the off-spinner geometry is shown in Fig. 3. The spinner plate is assumed to remain level and is located a height H, above the ground. A particle is assumed to leave the spinner plate with the total velocity given by Eqn (11). The forces acting on a particle after it leaves the plate are assumed to include only gravitational forces and aerodynamic drag. Lift forces due to particle rotation and local drag coefficient void fraction effects are neglected. From Newton s Second Law, a particle s equations of motion are: mx = mg C A ρ xx / 2 (12) D air where x is the particle s position, C D is the particle s drag coefficient, A is the particle s frontal-projected area, ρ air is the surrounding air density, and g is the gravitational acceleration vector (assumed to act in the downward vertical direction). To simplify the analysis, the drag coefficient is assumed to remain constant over the range of Reynolds numbers encountered by the particle. For spherical particles, the drag coefficient is nearly constant at 0.44 over the range of Reynolds numbers between 1000 and 200,000 (see for example, Fox & McDonald, 1999). However, for typical granular fertilisers that are more irregular in shape, the transition to a constant drag coefficient occurs at a much lower Reynolds number, typically around 50 (Lapple, 1951). For a typical fertiliser with a diameter of 2 mm traveling through air, the Reynolds number will be larger than 50 for particle velocities greater than 0.4 m/s. Eqns (2) and (3) predict that a sliding particle with a friction coefficient of 0.5 dropped at a radius of 5.0 cm will leave a 30.5 cm diameter spinner plate rotating at 540 rpm at a velocity of 10.1 m/s, well above 7

11 the critical transition velocity. Hence, a constant drag coefficient is a reasonable assumption. The initial conditions for solving Eqn (12) are: ( t = 0) = ( 0, H) x (13) ( = 0 ) = (,0) x t v 0 (14) where v 0 is the total particle velocity found from Eqn (11). The closed form solution to Eqn (12) subject to initial conditions in Eqns (13) and (14) for the time τ that a particle remains airborne (the particle will contact the ground when the vertical component of the position is zero) and the horizontal distance the particle travels L are given, respectively, by: ( H ) ( H ) τ = ln exp exp 2 1 (15) where: ( τ v ) L = ln 1+ 0 (16) τ τ Kg H HK K L LK v 0 v0 g 1 ρair CD A K 2 ρ V (17) The primed quantities in the previous equations are expressed in dimensionless terms. The quantity K is an aerodynamic resistance coefficient taking into account the particle s drag coefficient C D, frontal projected area A, volume V, and the ratio of the air to particle 8

12 densities ρ air /ρ (Grift et al., 1997). Eqns (15) and (16) have not been previously presented in the literature. 3. Experiments Experiments were performed to measure both on-spinner and off-spinner particle trajectories as well as material properties. Sixteen fertiliser materials, listed in Table 1, were used in the experiments. The objectives of the experiments were to determine typical fertilizer properties and determine how well the previously described models predict the trajectories for a much wider variety of materials than has been previously reported in the literature Material characterisation Measurements of the particle size distributions, densities, friction coefficients, and aerodynamic resistance coefficients were made for each of the fertilisers and are listed in Table 1. Details of the measurements are given in the following sections Particle shape Although a detailed analysis of the particle shapes was not made, a brief discussion of the general shapes is presented here. Five of the materials were roughly angular in shape, two were roughly spherical, and five had shapes that were somewhere between angular and roughly spherical. Urea was nearly spherical in shape while the AMS (ammonium 9

13 sulphate) standard and DAP (diammonium sulphate) had crystalline shapes. The pell lime was roughly spherical in shape but also had a large percentage of agglomerated particles Particle size distribution The particle size distributions and size guide numbers (SGNs - the median particle sieve diameter in hundredths of a millimeter as indicated by the Canadian Fertiliser Institute, 1986), were determined using a sieve analysis with aperture sizes ranging from 4.75 to 1.00 mm (Endecott s Ltd. ASTME:11 sieves). The SGNs are reported in Table 1 along with the diameters below which 84% and 16% (by mass) of the material was collected (corresponding to ±1 standard deviation). Twelve of the sixteen materials have an SGN within ±10% of the mean SGN value of 212, i.e mm. However, the uniformity of each material varies considerably as indicated by the large differences between the 84% and 16% values. The smallest SGN value of 123 corresponds to AMS Standard, which contained a large number of fines resulting, presumably, from the product s friability. The largest SGN of 292 corresponds to the pell lime, which contained a large percentage of agglomerated particles. Both of these materials also have the largest non-uniformity. Urea, a nearly spherical material, and the zinc micronutrient have the greatest uniformity of size. A typical size distribution plot is shown in Fig. 4 for the zinc micronutrient Particle density 10

14 The particle mass densities ρ, were determined by measuring the volume of water displaced in a graduated cylinder for a known mass of particles. The cylinder was tapped several times after the particles were poured into the cylinder in order to eliminate trapped air bubbles, and the displacement measurements were quickly made in order to avoid errors due to particle dissolution. More than twenty separate density measurements were made for each material to obtain an average value. The variations in each material s density measurements were less than 5% from the average. The particle densities for the sixteen materials are also reported in Table 1 and range from 1260 kg/m 3 for the urea to 2600 kg/m 3 for the limestone. Most of the material densities fall into two groups. Nine of the materials have a density within ±10% of 1910 kg/m 3 while five materials have densities within ±4% of 2510 kg/m Friction coefficient Since the friction coefficient µ, between the particle and the spinner vane and particle and spinner plate plays an important role in the spinner dynamics, measurements of the friction coefficient between the fertiliser materials and the spinner plate material (aluminum) were made for several of the fertilisers. A crude estimate of the friction coefficient was made by gluing fertiliser particles to a flat, wooden block, then placing the block on an aluminum plate and tilting it until the block first begins to slide. The tangent of the plate angle at which first movement occurs is the static friction coefficient. The friction coefficient measurements were only made for some of the materials. Since the measured values are all close to 0.5 with a large relative error of ±0.1, additional measurements were deemed unnecessary. While this measurement method is 11

15 not very accurate, it at least gives some estimate of the friction coefficients. Hofstee (1992) measured friction coefficients between aluminum and six different fertilizers at four different normal loads over a range of relative velocities between 1.0 to 21.0 m/s. His friction measurements ranged from approximately 0.33 to 0.44 over the range of tests, close to the values reported here. Note that Hofstee (1992) made measurements of the dynamic friction coefficient values rather than the static values. Hence, we expect out results to be larger than those reported by Hofstee (1992). Since the on-spinner dynamics utilize a dynamic friction coefficient, we expect our predicted results using the static value to give velocities that are smaller than the measured values. Note that spinner plates in practice are often coated with rust along with a variety of fertiliser dusts so a very accurate measurement of friction coefficient is unwarranted and instead reporting a range of possible friction coefficients is more reasonable Aerodynamic resistance coefficient The aerodynamic resistance coefficient K, of the particles plays an important role in the off-spinner dynamics as indicated in the previous analyses (Eqns (15) to (17)). Measurements of individual particle aerodynamic resistance coefficients can be difficult to make. Law and Collier (1973) designed an elutriator for determining the terminal velocity aerodynamic resistance coefficients of five fertiliser materials and three types of seed. Grift et al. (1997) described experiments utilising a 15.8 m tall, enclosed tube drop tower for determining the aerodynamic diameters of three different fertilisers. A drop tower apparatus was utilised here to measure the terminal velocity resistance coefficients, but instead of dropping particles in air which requires a large vertical distance before the 12

16 particle terminal velocity is reached, a 1.5 m tall, 7.6 cm inner diameter water column was used. There are two advantages to using water as the surrounding fluid rather than air. First, a particle reaches its terminal velocity in water in roughly 1% of the distance it takes in air. Second, the terminal velocity of the particle in water is approximately 5% of that in air and, as a result, the time required for the particle to pass a known distance can be easily measured using a hand-held stopwatch rather than requiring specialised electronic equipment. The primary disadvantage to using water as the surrounding fluid is that the terminal Reynolds number in water is much smaller than that in air. However, as long as the terminal Reynolds number is greater than approximately 50, the resistance coefficient will not vary considerably with Reynolds number (Lapple, 1951). The terminal aerodynamic resistance coefficient K can be determined by considering gravitational, drag, and buoyant forces acting on a particle: ρvg + C ρ v A + ρ Vg = (18) 1 2 D 2 w T w 0 where ρ is the particle density, V is the particle volume, g is the acceleration due to gravity, C D is the drag coefficient, ρ w is the water density, v T is the particle s terminal velocity, and A is the particle s frontal projected area. Solving for the aerodynamic resistance coefficient as defined in Eqn (17) gives: ρ K = ρw ρ g ρ v 1 air 2 T (19) where ρ air is the density of air and the terminal velocity is: v T H = T (20) 13

17 where H is the height the particle falls in the drop tower over time T. For the experiments performed here, the value for H was 1.5 m, and T ranged from 6.2 s to 19.2 s depending on the material tested. The error in measuring the transit time with a hand-held stopwatch was estimated at ±0.1 s. The terminal velocities range from m/s giving a range of Reynolds numbers from , well above the critical Reynolds number of 50. Twenty measurements were averaged for each fertiliser material to give the aerodynamic resistance coefficients reported in Table 1. The measured K values range between 0.12 m -1 and 0.45 m -1 with an average value of 0.24 m -1. These values are at the high end of the K values reported by Hofstee and Huisman (1990) and within the range of values measured by Law and Collier (1973) Particle motion on the spinner plate The spinner plate used in the experiments consisted of a flat 30.5 cm diameter aluminum plate with four 2.5 cm tall radial vanes spaced every 90 (Fig. 1). The plate was driven by a DC motor (Leeson Electric Corp. 12V DC) attached to a variable DC power supply (Nobatron DCR60). The rotational speed of the spinner plate was determined using a stroboscope (Nova-Strobe DA 115 KT). Individual fertiliser particles were dropped onto the plate from a height just greater than the vane height to minimise bouncing in the vertical direction. A drop radius of 26 mm was used in all of the experiments. The trajectory of a particle on the spinner plate was recorded using a high-speed video camera (RedLake MotionScope 1000S) operating at 500 frames/s. The particle radial 14

18 velocity was determined by measuring the radial distance the particle traveled between successive frames on the videotape. The distance measurements were made using a ruler attached to the rotating plate. The position measurements have an error of ±0.1 mm. The radial trajectories of four different materials (urea, DAP, coarse potash, and AMS Standard) at two nominally different spinner rotation speeds (~230 and ~540 rpm) were investigated. The data, presented in dimensionless form, are shown in Fig. 5a-h. The curves in the figures correspond to the analytical models discussed previously. Due to the camera s limited field of view, measurements could only be made at the outer half of the disc radius Particle motion off the spinner plate The distance particles are thrown from the spinner before contacting the ground was also measured using the previously described spinner plate apparatus. The surface of the fixed spinner plate was located 1.0 m above the ground. A row of 15 plastic collection trays, each with a depth of 8.9 cm and a rectangular planform area with dimensions 20.3 cm by 27.9 cm, was arranged in a single row emanating radially from the spinner center. The tray centers were spaced 30.5 cm apart. A plastic tarpaulin was spread on the floor beneath the trays to minimise the bouncing of particles from the floor into the trays. The entire spinner/tray experiment was located indoors to eliminate wind effects. The experiments proceeded by pouring the fertiliser material through a funnel with a 1.3 cm diameter exit from a drop height just greater than the height of the spinner vanes. The drop radius in the experiments was nominally 5.1 cm. After the material was 15

19 distributed from the spinner, the mass of the particles collected in each of the trays was measured using an electronic scale (Acculab VI-4800). Distribution experiments were conducted for all 16 materials at two spinner speeds (~540 and ~810 rpm). Three trials were conducted for each operating condition in order to verify repeatability of the experiments. A typical distribution result is shown in Fig. 6 for the granular boron distributed by the spinner operating at 810 rpm. The plot shows the percentage of material collected in each tray (compared to the mass collected in all of the trays) as a function of the distance from the spinner center. The total mass collected in all of the trays was typically 110 gms. As can be seen in the figure, the three trials produced very similar results. In order to compare the experimental and analytical results more easily, each tray s mass collected percentage is averaged over the three trials. The resulting averaged distribution curve is then fit to a Gaussian curve using a least squares approach in order to give a well-defined mean distribution distance and standard deviation. The Gaussian curve fit closely follows the averaged data except at points furthest from the spinner where the curve fit tends to under-predict the distribution of fertiliser. The mean distribution distances and standard deviations for all sixteen materials are presented in Fig. 7. There is a variation of less than 25% in the mean distances for a given spinner rotational speed. The mean distribution distances at 540 rpm vary between 2.7 m for AMS Standard and 3.4 m for pell lime. The distances for 810 rpm range between 4.0 m for the AMS Standard and 4.8 m for the pell lime. In general, the lower the material s aerodynamic resistance coefficient, the further the material travels, as expected. The distribution standard deviations are approximately 10-15% of the mean 16

20 distance values for 540 rpm and 10-20% of the mean for 810 rpm. Hence, the sixteen fertilisers tend to collect at a well-defined location that does not vary significantly between fertilisers. 4. Comparison of analyses and experiments 4.1. Particle motion on the spinner plate Figures 5a-h plot the experimentally measured radial velocities of fertiliser particles on the spinner plate along with the analytical curves given in Eqns (2) to (4) for pure sliding and Eqns (9) and (10) for pure rolling. A range of sliding friction coefficients is used in the analyses in order to show the effects of this parameter. The data generally lie between the pure sliding (using the experimentally measured friction coefficient) and pure rolling curves; however, the large uncertainty bars in the experimental data make drawing conclusions regarding friction coefficient and pure rolling trends difficult. Note that the radial velocity at the disc rim varies by 15 to 25% between the pure rolling and pure sliding (experimentally determined friction coefficient) cases and can vary by approximately 5% if the friction coefficient is varied by ±0.1. Hence, the analytical models perform reasonably well in predicting the on-spinner trajectories; however, assuming a range of friction conditions from pure sliding with a µ of 0.5 to pure rolling seems most appropriate Particle motion off the spinner plate 17

21 Figures 7 and 8 plot the mean distance and standard deviation data collected in the experiments, along with the distances predicted using Eqns (15) to (17) assuming an initial particle speed predicted by the on-spinner analyses (Eqns (2) to (4) and (9) and (10)) with pure sliding (µ = 0.5) and pure rolling conditions (refer to the discussion in the previous section for the justification for this range of values). There is significant variation between the analytical predictions for the distance travelled and the experimentally measured values. For the 540 rpm data, the models over-predict the distance traveled for all but two of the fertilisers. No consistent offset is observed in the 810 rpm data. To quantify the variation between the predictions and the experiments, we compare the relative difference between the predicted distance averaged between the pure slip and pure rolling cases, and the mean distance measured in the experiments. For the 540 rpm case, the variation ranges between 3% and 35% with an average variation of 15%. Agreement between the experiments and models is better for the 810 rpm data in which the variation ranges from less than 1% to 23% with an average of 11% error. A weak correlation between the error and aerodynamic resistance coefficient is observed with the error decreasing with increasing aerodynamic resistance coefficient. Equation (16) indicates that the horizontal distance traveled by a particle L is less sensitive to small variations in the initial particle speed v 0 as v 0 increases. This may account for the smaller error in the predictions at the larger spinner angular speed since inaccuracies in the on-spinner model have less of an effect on the horizontal distance travelled. 18

22 5. Conclusions A comprehensive experimental and analytical study was performed to investigate particle trajectories on and off a spinner spreader. Sixteen different granular fertilisers were used. Measurements of the significant material properties, on-spinner trajectories, and off-spinner distance distributions were made. Analytical models for the on-spinner and off-spinner trajectories were also presented and a comparison between the predictions and experimental results were made. The experimental data for on-spinner particle trajectories generally lie between the analytical models for the pure rolling and pure sliding conditions using a sliding friction coefficient of 0.5. Although the on-spinner motion is relatively sensitive to the friction coefficient for pure-sliding conditions, using a specific value for the friction coefficient seems unwarranted since accurate determination of this parameter is difficult, especially for materials and spinner plates used in practice. The average relative error between the experimental data for the horizontal distance traveled by the fertilizer materials after being thrown from the spinner plate and the combined on- and off-spinner models is 15% and 11% for the 540 rpm and 810 rpm spinner speeds, respectively. The error generally decreases with increasing aerodynamic resistance coefficient. Analysis of the off-spinner model indicates that larger spinner speeds will be less sensitive to variations in the particle speed leaving the spinner plate. These results suggest that the on- and off-spinner models presented here reasonably predict, on average, the distance the real fertiliser materials travel. However, these 19

23 models must be carefully applied for in-field conditions since other factors such as wind and terrain effects will significantly affect the particle spread patterns. 20

24 Acknowledgements The authors gratefully acknowledge the support of Royster-Clark, Inc. and the National Science Foundation for their support of this work.

25 References Canadian Fertilizer Institute (1986). The CFI guide of material selection for the production of quality granular blends, Ottawa, Ontario Cunningham F M (1963). Performance characteristics of bulk spreaders for granular fertilizer. Transactions of the ASAE, 6(2), Cunningham F M; Chao, E Y S (1967). Design relationships for centrifugal fertilizer distributors. Transactions of the ASAE, 10(1), Fox R W; McDonald A T (1999). Introduction to Fluid Mechanics, 5 th ed., Wiley Griffis C L; Ritter D W; Matthews E J (1983). Simulation of rotary spreader distribution patterns. Transactions of the ASAE, 26(1), Grift T E; Walker J T; Hofstee J W (1997). Aerodynamic properties of individual fertilizer particles. Transactions of the ASAE, 40(1), Hofstee J W; Huisman W (1990). Handling and spreading of fertilizers. Part 1: Physical properties of fertilizer in relation to particle motion. Journal of Agricultural Engineering Research, 47,

26 Hofstee J W (1992). Handling and spreading of fertilizers: Part 2, Physical properties of fertilizer, measuring methods and data. Journal of Agricultural Engineering Research, 53, Hofstee J W (1994). Handling and spreading of fertilizers: Part 3, Measurement of particle velocities and directions with ultrasonic transducers, theory, measurement system and experimental arrangements. Journal of Agricultural Engineering Research, 58(1), 1-16 Hofstee J W (1995). Handling and spreading of fertilizers: Part 5, The spinning disc type fertilizer spreader. Journal of Agricultural Engineering Research, 62(3), Inns F M; Reece A R (1962). The theory of the centrifugal distributor. II: Motion on the disc, off-centre feed. Journal of Agricultural Engineering Research, 7(4), Lapple C F (1951). Fluid and Particle Mechanics, Univ. Delaware Press Law S E; Collier J A (1973). Aerodynamic resistance coefficients of agricultural particulates determined by elutriation. Transactions of the ASAE, 16(5), Mennel R M; Reece A R (1963). The theory of the centrifugal distributor. III: Particle trajectories. Journal of Agricultural Engineering Research, 7(3), 78-84

27 Olieslagers R; Ramon H; De Baerdemaeker J (1996). Calculation of fertilizer distribution patterns from a spinning disc spreader by means of a simulation model. Journal of Agricultural Engineering Research, 63, Patterson D E; Reece A R (1962). The theory of the centrifugal distributor. I: Motion on the disc, near-centre feed. Journal of Agricultural Engineering Research, 7(3), Pitt R E; Farmer G S; Walker L P (1982). Approximating equations for rotary distributor spread patterns. Transactions of the ASAE, 25(6),

28 Tables Table 1 The sixteen fertilisers used in the study with their measured properties; SGN, median particle sieve diameter in mm 10-2 ; SGN 16, sixteen percent of the material has a particle size in mm 10-2 below SGN 16 ; SGN 84, eighty-four percent of the material has a particle size in mm 10-2 below SGN 84 Material name Material shape SGN, mm 10-2 SGN 16, mm 10-2 SGN 84, mm 10-2 Density (ρ), kg/m 3 Friction coefficient (µ) AMS standard Crystalline Granular kmag Angular Kmag Angular Coarse potash Crystalline Sulphur Angular Limestone filler Angular Triple Roughly spherical Muriate of potash Angular Boron micronutrient Roughly spherical to angular F472G Roughly spherical to angular Urea Spherical Zinc micronutrient Roughly spherical to angular DAP Roughly spherical Manganese micronutrient Roughly spherical to angular Roughly spherical to angular Pell lime Roughly spherical (agglomerates) Aerodynamic resistance coefficient (K), m -1 21

29 List of Figures Fig. 1. A photograph of the spinner plate apparatus used in the experiments Fig. 2. A schematic of the spinner vane geometry used in the analysis of the onspinner particle motion; the spinner rotation speed is ω, the particle drop radius is r 0, and the spinner radius is R Fig. 3. A schematic of the geometry used in the off-spinner analysis; the spinner disc radius is R, the rotational speed of the spinner plate is ω, the distance a particle travels from the spinner centre is L, the height of the spinner above the ground is H, and g is the gravitational acceleration vector Fig. 4. A typical particle size distribution plot of zinc micronutrient Fig. 5. On Spinner trajectories for particles with coefficient of friction µ of 0.4 and 0.6, for spinner rotation speed ω of ~230 rpm and ~540 rpm with a spinner disc radius R of cm and a particle dropping point at a radius of 2.6 cm for: (a), (b) and (c) DAP ; (d) and (e) coarse potash ; (f) and (g) AMS standard ; (h) urea ;, measured;, pure rolling; - -, µ;, µ - 0.1; , µ Fig. 6. A typical distance distribution plot; the data shown in the plot is for the boron micronutrient fertilizer at a spinner speed of 810 rpm and the particle drop radius r 0 of 5 cm;, trial 1;, trial 2;, trial 3;, average;, fit; mean distance of 4.38 m; standard deviation of 0.67 m Fig. 7. The mean distances ± one standard deviation travelled by the sixteen different fertilisers measured in the experiments and predicted by the analyses described in the text; the data is for a nominal spinner rotation speed of 540 rpm;, experiments;, pure sliding;, pure rolling Fig. 8. The mean distances ± one standard deviation travelled by the sixteen different fertilisers measured in the experiments and predicted by the analyses described in the text; the data is for a nominal spinner rotation speed of 810 rpm;, experiments;, pure sliding;, pure rolling 22

30 Figures Fig. 1. A photograph of the spinner plate apparatus used in the experiments 23

31 ω Fertiliser particle r Spinner vane r 0 r R Fig. 2. A schematic of the spinner vane geometry used in the analysis of the on-spinner particle motion; the spinner rotation speed is ω, the particle drop radius is r 0, and the spinner radius is R 24

32 y ω Spinner plate R H x Fertiliser particle g x L Fig. 3. A schematic of the geometry used in the off-spinner analysis; the spinner disc radius is R, the rotational speed of the spinner plate is ω, the distance a particle travels from the spinner centre is L, the height of the spinner above the ground is H, and g is the gravitational acceleration vector 25

33 SGN = 216 Proportion retained on sieve (by mass), % Sieve aperature size, mm 10-2 Fig. 4. A typical particle size distribution plot of zinc micronutrient

34 dimensionless radial velocity V r /(wr) dimensionless radial velocity V r /(wr) µ = 0.6 ω = 230 rpm dimensionless radial position r/r µ = 0.6 ω = 513 rpm dimensionless radial velocity V r /(wr) µ = 0.6 ω = 233 rpm dimensionless radial position r/r (a) (b) dimensionless radial position r/r dimensionless radial velocity V r /(wr) µ = 0.6 ω = 233 rpm (c) (d) dimensionless radial position r/r 27

35 dimensionless radial velocity V r /(wr) dimensionless radial velocity V r /(wr) µ = 0.6 ω = 531 rpm dimensionless radial position r/r µ = 0.4 ω = 543 rpm dimensionless radial velocity V r /(wr) µ = 0.4 ω = 251 rpm dimensionless radial position r/r (e) (f) dimensionless radial position r/r dimensionless radial velocity V r /(wr) µ = 0.4 ω = 230 rpm dimensionless radial position r/r (g) (h) Fig. 5. On Spinner trajectories for particles with coefficient of friction µ of 0.4 and 0.6, for spinner rotation speed ω of ~230 rpm and ~540 rpm with a spinner disc radius R of cm and a particle dropping point at a radius of 2.6 cm for: (a), (b) and (c) DAP ; (d) and (e) coarse potash ; (f) and (g) AMS standard ; (h) urea ;, measured;, pure rolling; - -, µ;, µ - 0.1; , µ

36 30 25 Proportion collected by mass, % Distance travelled, m Fig. 6. A typical distance distribution plot; the data shown in the plot is for the boron micronutrient fertilizer at a spinner speed of 810 rpm, and the particle drop radius r 0 of 5 cm;, trial 1;, trial 2;, trial 3;, average;, fit; mean distance of 4.38 m; standard deviation of 0.67 m 29

37 Distance travelled, m AMS standard Sulphur Limestone filler Kmag Granular kmag Coarse potash Zinc micronutrient F472G Triple Muriate of potash Manganese micronutrient Boron micronutrient DAP Urea Pell lime Fig. 7. The mean distances ± one standard deviation travelled by the sixteen different fertilisers measured in the experiments and predicted by the analyses described in the text; the data is for a nominal spinner rotation speed of 540 rpm;, experiments;, pure sliding;, pure rolling 30

38 Distance travelled, m AMS Standard Granular Kmag Kmag Coarse Potash Sulphur Limestone Filler Triple Muriate of Potash Boron Micronutrient F472G Urea Zinc Micronutrient DAP Manganese Micronutrient Pell Lime Fig. 8. The mean distances ± one standard deviation travelled by the sixteen different fertilisers measured in the experiments and predicted by the analyses described in the text; the data is for a nominal spinner rotation speed of 810 rpm;, experiments;, pure sliding;, pure rolling 31