Team:UMaryland/Model

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Model

Introduction

After selecting three designs based on background research and consultation with professor Ken Kiger at UMD, we wanted to create a model using a differential equation to describe the uptake of phosphorus within the cell. The parameters we would consider varying are chiefly the surface area and the volume occupied by the filters themselves. The model served in conjunction with our hand calculations to give us an appropriate time scale for how long the reactor would have to be run. Based on this model, we could also see which configuration of filter would be the most efficient in clearing the phosphorus from the water.

Introduction of the Three Designs

From consultation with our mentor and Dr. Ken Kiger form the Department of Mechanical Engineering, as well as background research of fermenter design, we chose 3 different filter designs to house the beads. The filter designs are shown below:

Twelve Columns

Singular circular filter

4-rectangle x-shaped filter

Important Factors to Consider

Next, we used a simplified model where we only wanted to consider the variations in Surface Area/ Volume of the tubes. A full list of parameters is listed below:

From this, we derived a differential equation that described the rate of phosphate uptake as a function of the concentration outside the cells, inside the cells, and the max phosphate that the filter could take.

Notice the quantity of 1 - Pin/Pmax, as Pin becomes close to Pmax, the rate of uptake decreases (diffusion limited)

Some of these values are derived from basic concentrations, which are stated below.

  • Concentration of Bacterial Cells = 0.67E8 cfu/mL
  • Max Phosphorous Per Cell = 4E-7 g/mL
  • Molar Mass of Phosphorous = 94.9714
  • Volume and Surface were calculated using traditional volume formulas of the given shapen (more specifics on the Engineering Page)

Code and Model

Thus, the main factor which would change in each of our designs was the max amount of phosphate which could be taken, as well as the available surface area for taking up the phosphate. To calculate these various areas and surface areas, we utilized hand-calculations, basing our dimensions on our CAD Model. We could repeat this procedure with many different types of designs, as the factors we are most concerned about are the surface area and volume.

Hand Calculations

Assumptions:

  1. The entire volume of the bucket is exactly 5 gallons and considered usable volume
  2. The filters are uniform in density
  3. Water is able to uniformly pass completely through the filters
  4. All filters are perfectly dimensioned and exact