(134c) Mathematical Modeling at the Food-Energy-Water Nexus

Maintaining a sustainable food supply for the foreseeable future without excessive environmental degradation will require novel engineering approaches for remediating the human disruption of the nitrogen cycle. Over the past several decades, food production has been able to keep pace with human population growth only thanks to the development of new high-yielding crop varieties grown with the help of fertilizers. Thus, vast amounts of energy are consumed every year to produce the approximately 200 Mt of ammonia needed to make the nitrogen (N) fertilizers used by farmers around the globe.

Unfortunately, more than half of the N fertilizer applied to a field is not available for plant growth due to losses caused by surface runoff, leaching into surface and ground water, or volatilization. As a result, the increased use of N as fertilizer has been linked to a variety of water pollution problems ranging from the expansion of the hypoxic zone in the Gulf of Mexico caused by eutrophication to the contamination of wells and groundwater with nitrate N. More importantly, denitrification reactions occurring in saturated soils convert nitrate to nitrous oxide, a major greenhouse gas and significant contributor to the depletion of atmospheric ozone

Several studies have reported that the addition of biochars to weathered tropical or temperate region soils decreased leaching losses of nutrients. In other cases, however, biochar amendments did not have the expected beneficial effects. Such contradictory results generated a lot of interest in the production of designer biocharsengineered and optimized to provide specific environmental performance. However, matching the properties of a biochar to a specific application is a daunting challenge. There are dozens of feedstocks from which biochars can be produced in multiple types of reactors under varying temperature and oxygen conditions, quickly leading to thousands of potential biochars with widely varying properties. Then, the performance of each of these biochars must be tested for a wide range of crops, soil types and climates.

Mathematical models derived from first principles can greatly speed up the process of selecting the most promising candidates for field-testing. By allowing us to isolate and analyze the interactions of key components of our soil-biochar-water system with the microbial populations carrying out the nitrification/denitrification reactions, mathematical modeling provides a systematic way for tuning the properties of a designer biochar to maximize its agricultural benefits. The amount of biochar applied to a field, for example, could be matched to the hydrodynamic characteristics (permeability, pore water velocity, soil type) of a specific field or to the amount of expected rainfall in order to achieve the desired nutrient retention effects. Or, we could select a feedstock and production method that would yield a biochar with optimal adsorption capacity and sorbate affinity.

The amendment of soils with biochar introduces a new stationary phase that necessitates an extension to the classical dual porosity models that considered convection and pore diffusion but only one adsorbent (stationary) phase. Biochars are highly porous materials with interconnected networks of pores that span multiple length scales: from sub-nanometer micropores to macropores with sizes of the order of 10 μm or larger. Moreover, the size and adsorption capacity of biochar particles may be significantly different than the corresponding properties of soil particles.

We will first discuss results from a dynamic convection-dispersion-adsorption model that considers nutrient (fertilizer) transport through beds with two porous adsorbent phases: soil and biochar. After a single fertilizer application, rain or irrigation water flows through the bed and nutrients diffuse in the water-filled pores of both stationary phases, adsorbing reversibly on their pore surfaces. The resulting system of PDE/DAEs is solved numerically using the method of lines. Simulation results will be presented to demonstrate that addition of biochar can effectively slow nutrient transport through the soil if the biochar/soil ratio and crucial biochar properties (like its adsorption capacity) are carefully matched to the soil properties (e.g. soil permeability) and the amount of rainfall or irrigation.

To further investigate the importance of biochar properties, we will focus on the interplay of external mass transfer, intraparticle diffusion and adsorption. We will analyze the dynamic behavior predicted by models considering surface diffusion of nutrients in addition to pore diffusion, pore structures with bimodal size distribution, and adsorption kinetics described by Freundlich or Langmuir isotherms. One of the more surprising findings here will be the sensitivity of the adsorption/desorption dynamics on the Biot number of the biochar particle. Biochar behaves like a â??controlled releaseâ? medium for adsorbed nutrients primarily because of the small Biot numbers that characterize the flow of rainwater through amended soils.

Finally, we will turn our attention to the problem of nitrogen transport and simultaneous reaction in biochar amended soils. Our equivalent continuum model will assume that ammonium-N is added to the amended soil as fertilizer, dissolves in water and is transported through the bed, while it is reversibly adsorbed on the two solid phases (soil and biochar). Ammonium is oxidized first to nitrite and then to nitrate by soil microbes (nitrification). The nitrate is uptaken by plant roots, immobilized (fixed) by soil microbes, transported along the bed through the convection/adsorption process described by the earlier models or converted to N₂, N₂O (a greenhouse gas) and NO_x (an atmospheric pollutant) by a different family of soil microbes (denitrification reactions). Preliminary simulation results will be presented and analyzed to show how key biochar properties and environmental variables influence the transport and uptake of nutrients, the loss of N fertilizer through leaching and the fluxes of nitrogen oxides.