Executive Summary Team 21
Statement of Problem: Determine the minimum height and wind speed that can be used to separate 30% - 40% of the paper that is in the falling column of material.
In this scenario, we will be using a commercial grade air knife to separate the paper from the cardboard. An air knife creates a high velocity air stream of which the falling materials have to travel through a shaft. The lighter material, the paper, is blown or separated from the falling material and is captured by the extraction hood and conveyed away. The heavier material, the cardboard, is unaffected by the incoming air stream and falls through the shaft onto a different conveyor belt. An air knife is a very common apparatus used in the recycling industry.
While most air knives use high velocity streams of air, we aim to minimize the speed of the air and the height from which the recyclable materials are dropped. In our model, the optimal height to drop the cardboard and paper is approximately .2794 meters above the fan. This height is ideal because the height will give the paper and cardboard just enough time to separate by about 11 inches, which allows for the paper and cardboard to fully rotate with the least amount of interference between it and the other materials. The ideal wind speed for our model is 6.155 m/s. This is the optimal wind speed because the wind is strong enough to change the direction of velocity for the paper but not enough to change the direction of velocity for the cardboard. The wind will only affect the paper while allowing the cardboard to fall to the bottom. In order for our model to work, we have made the following assumptions:
The distribution of paper and cardboard items are relatively uniform.
The dimensions of the paper and cardboard are the same (8.5 x 11 inch), where a piece of paper weighs 4.732 grams and a piece of cardboard weighs 22.680 grams.
The paper and cardboard initially fall from a horizontal position and are not crumpled or folded.
For the paper and cardboard to be considered separated, a piece of paper must move horizontally at least one meter from its initial dropping point.
The force due to air resistance is proportional to the speed and is applied in the direction opposite to motion.
In this model, we use a free body diagram to determine all the external forces acting upon the paper and cardboard. Through this free body diagram, we are able to derivetwo second order linear nonhomogeneous differential equations using Newton’s 2nd Law and then solve them using undermined coefficients.
In our equation, air resistance is proportional to velocity. The negative sign in front of the k constant is to indicate that air resistance opposes the direction of falling. Thus, the direction of falling is the positive direction. From Newton’s 2nd Law,
which can be rewritten as
Solving these equations gave us equations for the x and y positions of the falling materials, which we then used to find a minimum height and speed.
We chose to set the average mass of the paper as 4.732 grams and the average mass of the cardboard as 22.680 grams for an 8.5 x 11 inch sheet for both materials. The acceleration due to gravity is 9.807 m/s, and we estimated the proportionality constant k based on the shape of the materials and other environmental factors.
As the paper and cardboard fall down the main shaft of the air knife, both objects will fall at different velocities because of their different masseswhen we account for air resistance. Since cardboard weighs heavier than paper, the cardboard will fall faster down the shaft than the paper. After both objects fall for approximately1.88 seconds, there will be about 11 inches between the cardboard and paper to separate the two objects using a crosswind from the fan. The wind speed is determined by the amount of force it will take for the horizontal force of the air knife to overcome the vertical force of gravity and air resistance. However, the wind speed can only have enough force to have an effect on paper. If the wind speed is too strong, the cardboard might also be blown out of the main shaft and mix with the papers. In order to solve this, we had to set up an optimization problem based off the difference between the mass of the paper and the mass of the cardboard to find the minimum wind speed. Furthermore, in order to blow 30%-40% of the paper, we decided to use a pulsing system where the air fan will only run 30%-40% of the time to separate the paper. Air separation to sort paper from cardboard is an efficient method which is currently used, and there are other variations that are more efficient and accurate in separating recycling materials, one of which usinga high powered stream of air that blows lighter materials from the bottom. This model is likely faster and more accurate in separating the materials.