A magnetic braking and sensing technique for deceleration and recovery of moving non-magnetic metallic particles David Cheng, Akio Yoshinaka, and Lawrence Wu
Citation: AIP Advances 8, 056714 (2018); View online: https://doi.org/10.1063/1.5007303 View Table of Contents: http://aip.scitation.org/toc/adv/8/5 Published by the American Institute of Physics
AIP ADVANCES 8, 056714 (2018)
A magnetic braking and sensing technique for deceleration and recovery of moving non-magnetic metallic particles David Cheng,1,a Akio Yoshinaka,1 and Lawrence Wu2 1 Defence
Research and Development Canada - Suffield Research Centre, Medicine Hat, AB, Canada 2 Amtech Aeronautical Limited, 678 Prospect Drive S.W., Medicine Hat AB T1A 4C1, Canada (Presented 7 November 2017; received 2 October 2017; accepted 9 November 2017; published online 27 December 2017)
A magnetic braking and sensing technique developed as a potential alternative to assist with the non-contact deceleration and detection of explosively dispersed nonmagnetic metallic particles is discussed. In order to verify the feasibility of such a technique and gain an understanding of how the underlying forces scale with particle size and velocity, a study was conducted whereby an aluminum particle moving along a spatially varying but time-invariant magnetic field was modeled and the corresponding experiment performed. © 2017 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5007303 I. INTRODUCTION
In the study of energetic materials1,2 either contained in non-magnetic metallic casing or mixed with non-magnetic metallic particles, it is of interest to study the size and velocity distribution of the resulting explosively dispersed particles in the 10-1 – 100 mm range of sizes. Typical fragment recovery techniques involve the use of low density materials (e.g., low density fiberboard) as catching medium to recover these fragments. Unfortunately, particle deformation can still occur as a result of the impact with the capture medium, which interferes with post recovery examination. Furthermore, the need to separate particles from the capture medium makes the analysis very tedious and time consuming. Magnetic or eddy current braking was investigated as a potential alternative to assist with the non-contact deceleration of explosively dispersed non-magnetic metallic fragments. In order to verify the feasibility of such a technique and gain an understanding of how the underlying forces scale with particle size and velocity, a study was conducted whereby an aluminum particle moving along a spatially varying but time-invariant magnetic field was modeled and a magnetic pickup coil sensor was built and placed in the magnetic field to detect the aluminum particle as it travels through the field. When an aluminum particle enters the magnetic field of varying intensity, eddy current is induced within the particle. The particle is modeled as a current loop with a magnetic dipole moment perpendicular to the plane of the loop, creating another magnetic field around the particle which in turn induces an electromotive force (emf) inside the pickup coil. The strength of the emf in the coil is measured and used as indicators of the speed and momentum of the particle. Experiments were conducted on aluminum spheres of varying sizes using the proposed magnetic braking and sensing technique and the results show that the peak emf induced by a particle is clearly distinguishable from its surroundings, and the magnetic field required to induce it is of a reasonable magnitude. II. MODELS
The proposed magnetic braking and sensing system consists of an array of permanent magnets mounted on one side of the particle travel path, and pickup coil sensors placed on the opposite side to detect the moving aluminum particles (FIG. 1(a)). a
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FIG. 1. (a) Schematic of a magnetic braking and sensing system (b) Horizontal eddy current loop and magnetic dipole moment (c) Vertical eddy current loop and magnetic dipole moment.
When an aluminum particle enters the magnetic field, eddy current will be induced within the particle, and create another magnetic field around the particle. The particle is modeled as a current loop with a magnetic dipole moment perpendicular to the plane of the loop. Two orientations of the loop are examined: horizontal, where the magnetic dipole moment is perpendicular to the magnetic field (FIG. 1(b)), and vertical, where the magnetic dipole moment is parallel to the magnetic field (FIG. 1(c)). A. Horizontal eddy current loop
The magnetic dipole moment by the horizontal current loop is expressed as ~µz = −Ie · Ae zˆ , where Ie is the eddy current and Ae is the area of the magnetic dipole. The field strength along the axis of the pickup coil is given by µ0 3µz By = − · sin θ cos θ (1) 4π r 3 q t y0 where θ = arctan z(t) , r = y02 + z2 (t), z(t) = ∫ vdt or z(t) = v · t if the particle velocity is constant. The 0
total emf induced by the horizontal eddy current across the pickup coil of N turns is determined by ! dBy ∂By dr ∂By dθ dΦ ε = −N = −N · A = −NA · + · (2) dt dt ∂r dt ∂θ dt ∂B ∂B y0 v0 µ0 9µz µ0 3µz v z(t) q 0 where ∂ry = 4π · r 4 sin θ cos θ, ∂θy = − 4π · r 3 cos2 θ − sin2 θ , dr , and dθ dt = − 2 dt = y2 +z2 (t) . y0 +z2 (t)
0
B. Vertical eddy current loop
The magnetic dipole moment of the vertical eddy current loop is in line with the axis of the pickup coil, and is given by ~µy = −Ie · Ae yˆ . The magnetic field along the axis of the pickup coil can be calculated by the elliptical integral (R − z sin φ) dφ µ0 Ie R 2π By = − (3) 3/2 2 4π 0 R + r 2 − 2zR sin φ
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where R is the radius of the dipole loop. When r R, Eq. (3) can be simplified as µ0 2µy By = − · 3 3cos2 θ − 1 (4) 4π r q y02 + z2 (t). The emf induced by the vertical eddy current where θ = arctan z(t) , z = z − v t, and r = 0 0 y0 loop is determined by ! ∂By dr ∂By dθ ε = −NA · + · (5) ∂r dt ∂θ dt ∂By 3µ µy ∂B µy y0 v0 v z dr 0 0 2 q 0 where ∂ry = 3µ , and dθ 2π · r 4 3cos θ − 1 , ∂θ = π · r 3 cos θ sin θ, dt = − 2 dt = − y2 +z2 (t) . y0 +z2
0
C. Braking force
To simplify calculation of the eddy current induced within the particle as well as account for the resulting deceleration, the particle is modelled as a cube with sides equal to the diameter (2 × R) of the aforementioned current loop. The current density produced by the electromotive force is ~J = σ E ~ + ~v × ~B , where σ is the conductivity of the metallic particle, E ~ is the electric field intensity ~ induced within the particle, ~v is the particle velocity, and B is the magnetic field through which the particle is traveling. The electric field intensity in the direction of the y-axis, which is also in line with the axis of the pickup coil, can be obtained by performing integration using Coulomb’s law:3 " ! ! ! !# x−R x+R x+R x−R vB arctan − arctan + arctan − arctan = vB · δ(x, y) (6) Ey = 2π y+R y+R y−R y−R f g 1 x+R x+R x−R where δ(x, y) = 2π arctan x−R y+R − arctan y+R + arctan y−R − arctan y−R . The magnitude of the braking force can be calculated as the force component opposite to the particle velocity F = −σ
R R
Ey − vB BdV = −2σvB R
[1 − δ(x, y)]dxdy
2
V
(7)
−R −R
D. Motion equation
From (7), the motion equation along the x-axis direction (i.e., the direction of particle movement) can be established and solved. The particle position is derived as 1 x = x0 + v0 1 − e−λt (8) λ where x0 and v0 are the initial position and initial velocity of the particle, respectively, and λ = 2σB m
2R
R R
∫ ∫ [1 − δ(x, y)]dxdy is a coefficient dependent on the average magnetic field strength −R −R
and particle size. From (8), the particle velocity is given by v(t) = v0 e−λt , and the eddy current induced in the particle is given by Ie =
JdV =
V
R R σ(Ey − vB)dV = 2σvBR
V
[1 − δ(x, y)]dxdy =
mvλ B
(9)
−R −R
III. SIMULATIONS
To test the established emf models, numerical simulations were performed assuming a constant speed (1000 m/s) and eddy current loop (1 mA) with a radius of 50 microns. The pickup coil had 1000 turns and a radius of 5 cm and was placed midway of the permanent magnetic field of 2 meters in length. Meanwhile, the particle would travel in a straight line 5 cm away from the pickup coil. With these settings, both horizontal and vertical orientations of the current loop produced an emf on the scale of nanovolts. The vertical loop orientation (FIG. 2(a)) was found to produce roughly
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FIG. 2. (a) EMF induced by a vertical eddy current loop (b) EMF induced by a horizontal eddy current loop (c) EMF vs separation gap (d) Eddy current vs time (e) Velocity vs time (f) Induced emf sum of decelerating loop.
double the emf of the horizontal loop orientation (FIG. 2(b)). It was noticed in both scenarios that the peak emf occurring when the particle crossed the center of the coil can be clearly distinguished as the particle travelled through the magnetic field. The low emf magnitude may be offset in reality by the fact that the pickup coil would be exposed to a cloud of particles of various sizes, and superposition of emfs induced by individual particles would generate a much larger net emf in the coil. It was also discovered that the emf dropped very quickly within 1 mm of the separation gap, and would continuously approach zero with the gap increasing (FIG. 2(c)). To simulate the eddy current induced within the particle as well as account for the resulting deceleration, the particle was modeled as a cube of aluminum with a side length of 100 microns. The coil was moved to align with edge of the field with its properties unchanged (1000 turns and a 5 cm radius). The strength of the field was determined based on the condition that the particle is decelerated from 1000 m/s to 1 m/s across the field, which was calculated to be 0.222 T. The field length was reduced to 1 meter. The simulation results are shown in FIG. 2(d) through FIG. 2(f). The total time the particle spends in the field was calculated to be 6.915 ms. The average eddy current over this time period was 1.798 mA, relatively close to the previous estimate of 1 mA. The curves for eddy current and velocity
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were fitted to a 6th -order polynomial respectively. These polynomials were then used to adjust the model for induced emf, where eddy current and velocity were set constant over time. The peak emf of the adjusted model was much larger than in its previous counterpart because of the significantly higher eddy current seen when the particle first enters the field (FIG. 2(f)). IV. EXPERIMENTS
To verify the magnetic braking and sensing technique for deceleration and recovery of nonmagnetic metallic particles, a laboratory scale braking and sensing prototype system was built. The system consists of an array of permanent magnets mounted on backup steel bars, where each adjacent magnets have opposite magnetization directions. The magnet array length was 90 cm. A. Deceleration
In this experiment, single aluminum spheres of 3 mm, 6 mm, and 12.6 mm outer diameter (OD) are dropped and fall parallel to the magnet array. A 1mm hole was drilled through the diameter of the sphere in order to allow a nylon wire to guide the sphere during its descent. To detect the position and the time-of-arrival (TOA) of the falling spheres, laser diodes were aimed at optic fibers located at specific intervals along the sphere’s falling trajectory. Each fiber guided light to an amplified
FIG. 3. (a) Deceleration experiment setup (b) Comparison of deceleration effects on the aluminum spheres of different sizes (c) Pickup coil sensor experiment setup (d) Comparison of modelled emf and measured emf signals on the pickup coil sensor.
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photodiode where the drop in light intensity due to the shadow created by the moving sphere was time resolved on a digital oscilloscope (FIG. 3(a)). FIG. 3(b) shows the experimental results that compare deceleration effects on the aluminum spheres of different sizes. It is seen that the spheres eventually reach a constant velocity as gravitational and magnetic forces equilibrate. Further work is required to adapt the model to characterize the deceleration effects. B. Pickup coil sensing
To verify the emf models for recovery of the particle properties, a pickup coil sensor was built to have 1000 turns of copper wire and a radius of 5 cm (FIG. 3(c)). A 6 mm OD aluminum sphere was allowed to fall and traverse the magnetic field from the magnet array. The aluminum sphere freely falls along the aforementioned guiding string and enters the magnetic field region. The spacing between the center of the aluminum sphere to the edge of the pickup coil and to the surface of the magnets was identically set to 5 mm. The pickup coil was installed at the position where the outer edge of the coil is in line with the beginning of the magnetic field for higher eddy current when the particle first enters the magnetic field. The leads of the pickup coil sensor were connected to different oscilloscope channels, resulting in differential inputs to reject common mode noise. The voltage difference between these channels provides a measure of emf voltage across the pickup coil. FIG. 3(d) compares the emf signal computed by the analytical emf model and that measured by experiment. It is observed that the emf model reaches its maximum of 7.19 mv at 128.7 ms and its minimum of -5.59 mv at 139 ms, whereas the actual emf signal measured in the experiment has its maximum of 6 mv at 128.6 ms and its minimum of -4.4 mv at 137.7 ms. The peak voltages resulting from when the aluminum sphere enters and leaves the edges of the pickup coil at 118 ms and 152 ms are nearly absent in the analytical model because alternation of the magnetization direction is not accounted for in the model. The difference in the peak magnitudes of the modeled emf and the measured emf reflects the impacting factors that are not included in the analytical model, such as alternating magnetic flux directions, gravity, air drag, and guiding string friction. However, the experiments suggest that the emf models established in this paper provide a close estimate of the emf magnitudes induced by the eddy current in a metallic particle traversing the magnetic field, and the pickup coil sensor may provide an effective measure of the momentum of a cloud of particles. V. CONCLUSION
The magnitude of emf induced in a pick-up coil calculated by the numerical models shown in this paper is in the order of nanovolt level for an individual particle of size ranging between 10-1 – 100 mm moving at 1km/s. However, it is observed in the experiments that the emf is clearly detectable when a larger particle traverses through a magnetic field of similar amplitude. This result suggests that an aggregate cloud comprising a large number of smaller particles may generate a detectable signal with the pick-up coil technique. The field strength required to induce the emf is on the order of 10-1 T. The magnetic braking and sensing technique proposed in this paper provides a potential for improving the current techniques for fragment deceleration and recovery in explosive testing of energetic materials. Future research work will be conducted to relate particle momentum to the measured emf. Using multiple pickup coils that give data at different intervals in time and space may provide a feasible solution to this problem. ACKNOWLEDGMENTS
The Authors of this paper carried out this research on behalf of Her Majesty the Queen in right of Canada and the copyright in the work of this paper belongs to the Crown. 1 D.
Cheng and A. Yoshinaka, Review of Applied Physics 3, 44–48 (2014). Leadbetter, L. Donahue, R. C. Ripley, and F. Zhang, AIP Conference Proceedings 1195, 1035–1038 (2009). 3 W. H. Hayt and J. Buck, Engineering electromagnetics, 7th Edition (McGrw-Hill, New York, 2006). 2 J.
