Particle exit distributions

1. Particle exit distributions on a Galton board: Simulations and experiments J.G. Benito, G. Meglio, I.Ippolito, M. Re and A.M. Vidales Lab. de Ciencia de Superficies y Medios Porosos, UN San Luis, Grupo de Medios Porosos, Fac. Ing. Univ. Buenos Aires FAMAF, Universidad Nacional de Córdoba. CONICET, Argentina.

2. Introduction Obstacles arranged like periodic lattices are often used to study transport and diffusion in many phenomena ranging from crystals to grain mixing. Galton Board particle mixer has been developed and successfully proved Stillremains many questions......Presence of walls? this situation is common in industrial mixing problems. Aim of the present work is to study the influence of walls in the percolation of small disks through a vertical Galton board

3. Our study will basically contain three parts: experiments, numerical simulations and a theoretical approach. Measure exit distributions of particles at the bottom of the board as a function of the separation between lateral walls. Large range of size parameters inspected using simulations A theoretical model will confirm both experiments and simulations We will find a criteria to select the dimensions of a Galton board for practical purposes and applications.

4. Experiments small gap of 1.5 mm for particles to fall down 120 cm Discs diameter = 8 mm, materials: rubber, aluminium and lead Obstacles  styrene discs, (4 mm diam.) glued onto the wooden wall hexagonal network with lattice spacing of 2.4 cm. number of rows =46 two lateral PVC rods W = 70, 60, 50, 38, 26 and 16 measured in number of bins. 80 cm colector

5. A L U M I N I U M

6. Dependence on the material (○) lead, (□) aluminium and (∆) rubber discs  increases up to W  40 and then, tends to a constant. Up to W  40,  values  corresponding to uniform distributions for W > 40, the influence of the walls becomes less significant  distributions are quite normal  tends to a constant. The same variation is found for all kinds of materials used in experiments  depends on the geometry of the system.

7. Simulations A disk falls down by gravity until it encounters a pin. The small disk will roll over the pin to the right or to the left. Then, we have to decide which will be the new x-position of the particle (introduce bouncing effects) We are not interested here in the dynamics of the problem. Thus, we just choose a random number between 0 and 1 (ran(0,1)) and put the particle at the corresponding x-position given by:

8. We performed simulations for different values of delta in order to introduce a pseudo-bouncing effect, i.e., the smaller the value for delta, the smaller the hardness of the percolating particle. This simple approach will suffice for our purposes  to study the "re-orientation" of the disks in their way down to the exit due to the presence of lateral walls 105 equal experiments. This exit distribution function is fitted with a suitable curve in order to characterize the final distribution of the particles through its dispersion s. At the exit  particles collected  histograms  exit distributions.

9. Dispersion for normal distrib. Number of particles that touch the walls at least once. Characterization of such "tails" is not straightforward

10. We did simulations for different number of rows, scanning all the separations values to build The extent of the crossover depends on the number of rows of the board and the limiting values for large W (Gaussian behaviour) depend on the number of rows because of diffusion effects.The inception point of each curve on the straight line also depends on the height of the board. What is the relationship between the crossover found and the size of the Galton board?

11. To answer this question we first have to define, following some criterion, the critical value for W (WC) at which the exit distribution might start to be considered as uniform. Fitting with a second order polynomial function if one is searching for a uniform exit distribution of particles flowing down through a Galton board of Nrow, one needs to have, at least, a number of columns proportional to

12. This way is pretty much dependent on the initial criterion for choosing WC Theoretical approximated Model We consider the one dimensional diffusion of a particle in the finite range -L/2 ≤ x ≤ L/2, to approximate the solution of our discrete problem. Let us denote by P(x,t) the conditional probability density of finding the particle between x and x+dx given that it started at the centre of the interval [-L/2, L/2]. Solving the diffusion equation with reflecting boundary conditions and the initial condition , we obtain the solution App. to the discret problem

13. Let us consider a RW on a finite one dimensional lattice of spacing a. Assuming W+1 lattice sites the size of the space domain is L=aW. Choosing the coordinate origin at the centre of the lattice the walker position is x=a.s, with s an integer number with values –W/2 < s < W/2. Assuming reflecting boundary conditions, the probability of finding the walker at lattice site s after N steps is approximated by: criterion for any system size. W  WC N will represent the number of rows.

14. Finally, we did a similar set of simulations but with a different “bouncing” parameter. We chose a lower value for d  dl . A board with a fixed number of rows will need closer walls to distribute uniformly particles with lower bouncing coefficient. If the coefficient is higher, percolating particles explore even further regions of the board, noting the presence of walls with greater likelihood. Thus, the crossover region from Gaussian-like to uniform-like behaviour shows up earlier  higher values for WC.

15. Conclusion Existence of a crossover from a Gaussian to uniform behaviour. The relationship between the number of rows and the number of columns of the board was parabolic. A discrete theoretical approximation helped to set up a criterion to discern among uniform and Gaussian behaviour. Given the dimensions of a Galton board to be used as a scatter device, one can find the departure from uniform exit behaviour for the dispersed particles, and vice-versa. Finally, simulations for a different bouncing parameter showed that the crossover to uniform exit distribution behaviour shows up at greater WC as bouncing is increased. Influence of walls in a Galton scatter has been well characterized. Future efforts will focus on the problem of uniform mixing of two species of particles with variable wall separation.