SCUDEM 2018 Problem A Sorting Recyclables Lawrence Choi

SCUDEM 2018 Problem A Sorting Recyclables Lawrence Choi, Estelle Lee, Logan Tabor

The Problem The question was to explore whether or not a simple process can be developed to help separate a large percentage of materials, specifically paper and cardboard materials. - The device tested includes materials being dropped from a great height and a fan that will blow air across the stream of material. Determine the minimal height and wind speed needed to separate 3040% of the paper that is falling in the stream of material.

Assumptions - The distribution of paper and cardboard items are relatively uniform More than 30% of all recycled paper products are standard sheets of paper Paper can be considered separated when it is 4 meters away from the original landing area The fan can be turned on during the procedure without hindering the wind velocity

Determining Terminal Velocity - - In order to decrease the chance of other materials being separated with the paper, we can determine the terminal velocity of each item so that they separate during the fall - This means that when we solve for wind speed, the wind will only affect the falling paper rather than all the materials To determine the terminal velocity, we need to calculate the acceleration using the force of gravity and the force of air resistance, leaving us with the equation: Fg - Fa =a m

Determining Terminal Velocity - From this equation, we can find the velocity using an exponential growth differential equation By using Fg = mg and Fa = (air density)(drag)(area of object)* v/2, in which air density, drag, and area are constant for each material, we came up with these three equations mg v = k (1 - e^(-kt/m)) Paper: 14. 77911 t) v =. 663098 (1 - e^- Plastic: . 214694 t) v = 45. 646468 (1 - e^- Metal: v = 235. 600471 (1 - e^-

Determining Terminal Velocity - - For each of these equations, we took the limit as t approaches infinity to determine the terminal velocity of each material By examining the graphs of these equations, we determined that given enough height, the terminal velocity for each material would cause them to become separated at a certain height, meaning that we could use wind to push the paper and ultimately separate it from the other materials before reaching the bottom

Determining Wind Speed From Fan - - One of our assumptions is that the paper can be considered separated when 4 meters away from the original landing area. This meant that we had to find the necessary wind velocity and the amount of time needed at the specified wind velocity to blow the paper 4 meters away from its initial spot. To do this we used the following equations: Mass * acceleration=((air density)(area of object)(wind velocity))/2 distance=initial velocity*time +. 5(acceleration)(time)^2

Determining Wind Speed From Fan - - We researched different fans that could be used in this scenario, and the wind speeds that those fans could produce. This gave us wind speeds to plug into the equations and observe which seemed the most realistic. We decided to use a wind velocity of 7. 2 m/sec This gave us a time of. 37 seconds needed to push the paper 4 meters away from the initial starting point.

Is This a Viable Option? We have determined that this method is viable in real life, and the optimal height for the recyclable materials to be dropped from is about 1. 5 meters. However, to allow for a more efficient system, we have decided that if we were to construct this, we would drop the materials from up to 10 meters high. This would allow for the materials to separate on their own due to their differing terminal velocities, and permits a greater separation between the paper and other materials.

Additional Issue: Which aspect of your model results in the largest difference in sorting quality if that aspect undergoes a small change?

Additional Issue A small change in the height at which the material is dropped would change the outcome of our models. Our model accounts for the different materials separating at the same rate regardless of the overall mass, as we assume that the mass and the area would change proportionally. Because of this, our model would have to factor in the change in air resistance resulting from the additional materials, and a change in height could mean that the materials do not completely separate before using the fan, causing other objects to potentially be separated with the paper.

Comparison: A small change in the height would result in a bigger difference in comparison to a small change in wind speed because the height changes the rate and which the materials separate, while the wind speed only affects the paper in our model. A change in wind speed would create negligible affects regarding separation of the materials. However, a change in the height would create a bigger effect because this impacts the separation of materials in the air, which is essential to our model.

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