The Average Mesquite Bean
All calculations that are done were completed using the average of three mesquite beans.
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Required Speed to Knock down a Mesquite bean
In order to better understand what we would be getting ourselves into with this new method of harvesting we needed to set up some goals for ourselves. The main goal we needed to quantify was how fast does the working fluid (which is air) would have to be to knock down a mesquite bean. To do this some assumptions had to be made.
- A mesquite bean on a tree can be modeled as a cantilever beam with a distributed load that is produced by the wind coming out of a nozzle.
- The primary force being analyzed is the bending moment produced by the wind that will produce a stress that will be greater than the modulus of rupture (MOR). The modulus of rupture for mesquite beans is 16750 psi
- The weight of the bean itself is negligible. This itself would provide a normal stress which in itself is negligible with the forces we are analyzing.
- The single point of failure of a mesquite bean is the stem. This is the only thing we have to break for the bean to fall.
Mesquite Beans on a tree
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Mesquite stem modeled as a cantilever beam
With the assumption of the cantilever beam we can determine the bending moment required to break the steam using the average mesquite bean values as well as the MOR discussed above.
1. Finding the critical moment needed to break the stem
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With the critical moment obtained, the force of wind required can be solved for. This in turn can be then turned into the required wind speed needed to break the stem by looking at the cross-sectional area of the beam that the force of wind will be working on.
2. Calculating the force of wind needed to break the stem
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 Coefficient of Drag Formula
Using the average mesquite bean dimensions we can find an approximate coefficient of drag of 1.34. We assume that the bean can be modeled after a flat plate.
3. Finding the velocity of the wind needed to break the stem
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Through these calculations we discovered the required speed that the air would need to travel would be 19.52 miles per hour. This would serve as a baseline for the project and as a goal we would try to hit with our simulations and testing.
Calculations for Iteration 3
With iterations 1 and 2 serving as a proof of concept for the project, the team was ready to investigate the possibility of a small scale prototype in order to further our understanding of the system and the potential it had without investing in components that are more expensive. Looking at the experimental design there are three points of interest at which the wind speed needs to be calculated. The points are the fan exit, duct exit, and nozzle exit.
Known values of iteration 3
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The values of the fan and duct are shown to the left. This would allow us to predict the behavior of the air at the exit of the fan, as well as the exit of the duct.
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With a theoretical exit velocity of 31.2mph, the theoretical exit velocity of the duct can be calculated taking head loss into account.
1. Friction factor of the system
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2. Finding the head loss of the system
3. Calculations for ideal theoretical exit velocity
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The value of 28.16mph is above our needed velocity of 19.52mph, but as the air is traveling over longer distances, the strength of the air is reduced as it dissipates into the atmosphere. In order to mitigate this effect, and allow the air to travel a longer distance, the idea of attaching a cone would be implemented. Simultaneously while exploring the cones and their effect on air velocity, research went into the final parts the system would need. This would be the beginning of iteration 4 while also exploring nozzle designs and their effect on iteration 3. The goal would be to scale up the most optimal nozzle from iteration three and utilize that on iteration 4 as a starting point for experimentation.
Calculations for Iteration 4
With iteration three complete, iteration 4 was ran with a larger blower and longer duct in order to increase the output velocity of the system with longer lengths. This would allow ease of use for the end user and would help them harvest the complex geometries of the mesquite tree. These parts where sourced while testing of nozzles where being done on iteration 3. The output velocity was then utilized to run simulations on a theoretical nozzle that will fit a larger duct.
Known Values of Iteration 4
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The values of the fan and duct are shown to the left. This would allow us to predict the behavior of the air at the exit of the fan, as well as the exit of the duct.
Theoretical Fan exit Velocity
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With a theoretical exit velocity of 43.41mph, the theoretical exit velocity of the duct can be calculated taking head loss into account.
1. Finding the friction factor of the system
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2. Finding the head loss of the system
3. Calculations for ideal theoretical exit velocity
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This theoretical exit velocity served as a grounds for our fluid dynamic simulations in trying to find the optimal nozzle for our system.
