by Isaias Peraza
In the next lines, I will share with you my experiences teaching the third edition of a mathematical modeling workshop in my home state, Yucatan, and my participation in a renewable energy conference in the same location. The first two editions of the modeling workshop targeted graduate students in Environmental Engineering back on May of this year. This time, I taught Bachelor of Science students from the Technological Institute of Merida. The modeling workshop was one of many events to celebrate the annual week of biochemical, chemical and environmental engineering students hosted by the Chemical and Biochemical Engineering Department; an important occasion where they invite researchers and professors from local research centers but also, from other states of the country, to deliver various talks and where the most popular student activities are the workshops. I had 25 students to work with and the workshop cost them five U.S. dollars!
We started our 1st of four day workshop by asking, why is mathematical modeling important in environmental engineering? We concluded that math is used every single day in lab life and research! Scientists use math every time they design and set up experiments and reactions.
We discussed the relationship between field observations, laboratory studies, and mathematical modeling. Environmental studies and remediation procedures are necessary when problems emerge at a contamination site. However, they tend to be costly. They are instrumental in characterizing the extent of the environmental issue, identifying its bounds, and evaluating whether proposed remediation strategies are beneficial at a specific site. Laboratory studies, on the other hand, may only be useful in understanding the basic principles governing an issue at a micro or molecular scale. Findings and knowledge gained at this scale may experience significant problems when expanded to field-scale applications. Nevertheless, laboratory studies are beneficial for both solving problems and for understanding micro scale issues at various stages of environmental pollution investigations and remediation.
Modeling complements these two scientific approaches. Like laboratory studies, modeling can theoretically solve problems, but is more adept at controlling variables. Workshop activities progressed to simple model creation from an environmental system very familiar to all of us, an aquarium! By making appropriate assumptions, we formulated and solved a system of three time-dependent Ordinary Differential Equations (ODEs) in a batch mode.
Later in the workshop, I showed the students how to model a classic example in enzyme kinetics, the Michaelis-Menten (M-M) model, in which an enzyme consumes a substrate to generate a product, with a Substrate-Enzyme Complex. S→P; E+S↔ES→E+P
The main reason for working with the (M-M) model was to give the students an introduction to Monod kinetics, which addresses bacterial growth and its importance in biogeochemical models of transport and reactions with contaminants in the environment.
Both systems of equations, the (M-M) model and the aquarium, were modeled creating and solving ODEs, and using the solvers ode45 and ode15s that are part of MATLAB. Such solvers use the Runge-Kutta and the Order Variable methods to solve non- and stiff-differential equations, respectively.
The following days of the workshop, we ventured into the world of transport, to understand the essential features of the fate of environmental contaminants. I explained the concept of flow velocity and how can we model it using simplified expressions obtained from the general Navier-Stokes equation, since contaminant flow is a crucial topic in many environmental sciences. Flow is the carrier of advective transport. Biogeochemical species are transported by flow through environmental compartments.
Often, flow transport is the fastest process by which a potentially hazardous chemical species disperses from a source and reaches a sensitive region.
I gave the students an example of a repository for underground oil spills since this is a common problem in Yucatan. The primary safety problem with these waste disposal sites is concerned with the identification of flow. There is a danger that, in the long run, hazardous substances will find a subsurface flow path which takes them up to the surface.
Our workshop activities continued by implementing a numerical method for solving partial differential equations (the finite differences method). The objective was to give the students a first glance at the numerical technique that they can use to solve transport equations. For practice, we solved a popular problem of transport and biological reactions present in the literature for sulphate reduction. The model proposed by Boudreau y Westrich (1984) deals with the system:
In this model, bacteria utilize organic carbon as an electron donor. It is assumed that the rate of bacterial consumption is directly proportional to the concentration of utilizable organic matter.’ Taking into account the sedimentation velocity (w), which is derived from the burial rate, the model solves the differential equation for the steady state of organic matter in the solid phase.
To solve this and other models presented in the workshop, we wrote programs using MATLAB to deal with . We used the solvers: bvp4c and bvp5c.
At the end of every day, the students had to do homework related to the workshop, including written modeling questions and solving differential equations thought MATLAB programming.
Finally, on the last day of the workshop, we participated in a group discussion about the importance of modeling in environmental engineering. Students were provided with scientific papers related to modeling and videos about simulations to motivate them to continue working on modeling throughout their careers.
Also, during this trip, I attended a conference, the second edition of the Regional Energy Use and Renewable Energies Conference. This edition was very special since it was organized almost entirely by renewable and chemical engineering bachelor’s students. Here I got to hear and learn the status of energy-related issues of Yucatan, Mexico. Every day of the three days conference, there were two big plenary lectures and several research-related lectures and informative talks (mainly representatives of the different energy agencies of Mexico). The conference brought together researchers, students, and representatives from the energy sector and governmental agencies to discuss the current problems that the Mexican Southwest is facing and suggested possible solutions. I got to know about several other mega-projects like wind-farms and photovoltaic plants, and the potential social, economic and environmental problems that could come along with them. To learn more about this, please read the blog: https://environmentalbiotechnology.org/2018/06/21/the-not-that-green-business-of-wind-energy-in-yucatan-mexico/.
The second day of the conference I got to present my talk: ‘’Mathematical modeling in Microbial Fuel Cells and its Biofilms.’’ Only one other student presented research related to wastewater and bioenergy generation. The work dealt with the problem of methane production from pig manure wastewaters. During my presentation, I received a couple of questions from the audience, but I enjoyed an extended discussion afterward, in the poster session. Overall, my participation was very fruitful and the connections that I created will surely bring me possible collaborations in my future research work.
Isa Peraza is a 3rd year Ph.D. student in Civil, Environmental and Sustainable Engineering in the Ira A Fulton Schools of Engineering.