Bioreactor
Overview
We aim to model the growth of our E. coli within our bioreactor. We simulated the growth environment using MATLAB under assumed ideal conditions for E. coli growth and an exponential feeding cycle. This was accomplished using equations for growth kinetics and exponential feeding that would accurately model the growth of the bacteria inside of the bioreactor. Our goal is to add theoretical support to the function of the bioreactor, while simultaneously testing which conditions are most favorable for E. coli growth.
Process
In order to model the bioreactor’s cell growth, we used Monod’s equation [1] for modelling the growth of the bacteria:
We assume that the bioreactor provides ideal conditions for the E. coli: a temperature of 37º C and a pH between 6.5-7.5. Ultimately, the log phase of bacterial growth was modeled. This was done through exponential feeding, ensuring that the E. coli maintained log phase for the entirety of its time in the bioreactor growth chambers. The flow rate for new feed entering (as well as old feed exiting) the bioreactor was 0.460 L/min, which corresponded to the maximum speed of the motors used in the bioreactor design. Additionally, the carrying capacity was plotted against time, yielding a modeled value of 8000000 cells/L after one hour of growth in the bioreactor. This was done by rearranging the following equation [1] to solve for K:
The E. coli cell growth modeling was written using MATLAB software.
Results
Figure 1. Graph of the Growth Rate of E. coli within one hour.
From this graph, we can see that the carrying capacity came out to be 800000L cells/L after 1 hour. We also found that the growth rate models an exponential growth of E. coli cells, which relates to the log phase of cell growth. In addition, we had an initial number of cells set to 1000000 and can see that more cells continued to grow as time progressed. Therefore, using this model, we can quite accurately keep the cells in log phase for the entirety of the experiment, as long as we continue to keep an eye on the containers and ensure there will be no spillage.
Code can be found here.
Resources
- 1. Zelić, B., Vasić-Rački, Đ., Wandrey, C., & Takors, R. (2004). Modeling of the pyruvate production with Escherichia coli in a fed-batch bioreactor. Bioprocess and Biosystems Engineering, 26(4), 249–258. https://doi.org/10.1007/s00449-004-0358-0
- 2. Hixon, M. A. (2008, August 6). Carrying Capacity. ScienceDirect. Retrieved October 10, 2021, from https://www.sciencedirect.com/science/article/pii/B9780080454054004687
Fluids
Overview
Our fluid modeling aims to predict the behavior of our hydrogel gradient creation before crosslinking, as the two hydrogel types are mixed, and ensure that a proper gradient is able to be created. We use ANSYS fluids modeling software to model the flow of hydrogel through our novel gradient machine, with particular focus on how the two kinds of hydrogels are mixed together in the gradient machine before they are output and how this changes at different points in the extrusion process (i.e. when their input velocities are changing at different timesteps).
Process
The fluid models were created using the ANSYS software. The final product went through multiple rounds of iterations as we tried to discover what was feasible and what kind of modeling we wanted to do. We decided to use exact dimensions to model the flow of fluid inside of the gradient mixer at different points in the gradient making process, therefore having different ratios of inlet velocities and seeing how this affects the proportion of our two hydrogel types entering and exiting the gradient maker and how they mix. Using ANSYS, we created models of the velocity, pressure, and temperature within the gradient mixer. We used temperature because it is an analog of concentration and by modeling temperature, we can get a sense of how the fluids have mixed within the gradient mixer. Once we had the standard model (assuming laminar flow), we changed the inlet and outlet velocities so the sum would add up to .05 ft/s which would model the different stages in the gradient maker process. From this, we made another model using turbulent flow, making assumptions which are explained below. Additionally, we used results from a scientific study conducted by Saroja Ramanujan called Diffusion and Convection in Collagen Gels: Implications for Transport in the Tumor Interstitium [1] to estimate the diffusivity of these collagen gels. The diffusivity of these collagen gels was low because collagen is a large molecule and the time scale of this process is relatively short. As previously discussed, we used temperature as an analog to model different concentrations of collagen protein being pushed through the gradient mixer, and set the model’s initial temperature of the hydrogel in the bottom inlet to 1ºC and the temperature of the hydrogel in the top inlet to 0ºC to most effectively see the mixing of these hydrogels together.
Assumptions
- Viscosity : 0.00173 [kg/(m*s)] (viscosity between water and oil)
- Velocity top 0.04 ft/s (75%)
- Velocity bottom 0.01 ft/s (25%)
- Diameter: 0.071 in (measured directly)
- Density: 1,000 kg/m^3 (hydrogel density similar to water)
- Cp (specific heat): 1,000,000 [J/(K*kg)]
- Thermal conductivity:
- Reynolds number: 12.709 (using max velocity)
- We used the k-epsilon turbulence model
Results
From the results shown below, we can see that the distribution of the velocities, pressure, and concentration of the hydrogels are as expected for the different velocities of the hydrogels entering the gradient mixer. For hydrogel 1, entering at a greater velocity means a greater composition of that mixture of the hydrogel at the end. There is great pressure at the inlet with the hydrogel with the greater velocity as well. This is good because, with the pressure at the inlet, this keeps fluid moving through the gradient maker and allows the liquids to mix. The proportionality of the pressure provides insight as to the velocities of the hydrogel and which hydrogel is being moved along more than the other. Also, from a design perspective, the pressure is not great enough to affect the stability or integrity of the gradient mixer which is positive. From our current model with the laminar flow, we can see that the hydrogel mixture is almost homogenous by the time the hydrogel reaches the outlet. Using turbulent flow, we can see that this produces a fully mixed hydrogel by the time the mixer reaches the outlet, so this model would be preferable in order to produce a fully homogeneous hydrogel. To determine if our model meets the criteria to use a turbulent flow model, we calculated the Reynold's number. If the Reynold’s number is above 2,000, a turbulent flow model is warranted. However, anything underneath this number warrants a laminar flow model. Calculating the result, we got a Reynold’s number of 12.709 which is under the threshold to use the turbulent flow model. However, if we increase the velocity of the incoming hydrogels and/or the diameter of the tubes used to usher the hydrogels into the gradient mixer to create a Reynold’s number of 2,000 or over, we can use the turbulent model to create a homogenous mixture of the hydrogels by the time the mixture reaches the outlet.
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| Velocity of 5%-95% Hydrogel | Pressure of 5%-95% Hydrogel | Temperature of 5%-95% Hydrogel |
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| Velocity of 25%-75% Hydrogel | Pressure of 25%-75% Hydrogel | Temperature of 25%-75% Hydrogel |
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| Velocity of 50%-50% Hydrogel | Pressure of 50%-50% Hydrogel | Temperature of 50%-50% Hydrogel |
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| Velocity of 75%-25% Hydrogel | Pressure of 75%-25% Hydrogel | Temperature of 75%-25% Hydrogel |
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| Velocity of 95%-5% Hydrogel | Pressure of 95%-5% Hydrogel | Temperature of 95%-5% Hydrogel |
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| Turbulent Velocity of 25%-75% Hydrogel | Turbulent Pressure of 25%-75% Hydrogel | Turbulent Temperature of 25%-75% Hydrogel |
The Reynolds Number Equation was taken from Marine Propellers and Propulsion (Fourth Edition), 2019 [2].
Accreditations: Dr. Peter Doerschuk (BS, MS, PhD, and MD) of Cornell University and Parker Dean, a senior undergraduate student majoring in Biomedical Engineering at Cornell University who is also a teaching assistant for BME 2000/ENGRD 2202, were integral pieces in helping with the fluid modeling and using ANSYS. Professor Doerschuk assisted with the planning for the modeling and provided basic, yet essential, information about fluid mechanics. Parker Dean assisted with the modeling in ANSYS itself and served as an advisor and mentor in the fluid modeling. To both of these amazing people, we offer our greatest thanks for their generous and thoughtful assistance on this project.
Resources
- 1. Saroja Ramanujan, Alain Pluen, Trevor D. McKee, Edward B. Brown, Yves Boucher, Rakesh K. Jain, Diffusion and Convection in Collagen Gels: Implications for Transport in the Tumor Interstitium, Biophysical Journal, Volume 83, Issue 3, 2002, Pages 1650-1660, ISSN 0006-3495, https://doi.org/10.1016/S0006-3495(02)73933-7.
- 2. J.S. Carlton, Chapter 4 - The Propeller Environment, Editor(s): J.S. Carlton, Marine Propellers and Propulsion (Fourth Edition), Butterworth-Heinemann, 2019, Pages 47-57, ISBN 9780081003664, https://doi.org/10.1016/B978-0-08-100366-4.00004-3.