Mathematics of carbon footprints

Summary: A carbon footprint is a mathematical calculation of a person’s or a community’s total emission of greenhouse gases per year.

Carbon footprint is intended to be a measure of the ecological impact of people or events. It is a calculation of total emission of greenhouse gases, typically carbon dioxide, and is often stated in units of tons per year. There is no universal mathematical method or agreed-upon set of variables that are used to calculate carbon footprint, though scientists and mathematicians estimate carbon footprints for individuals, companies, and nations. Many calculators are available on the Internet that take into account factors like the number of miles a person drives or flies, whether or not he or she uses energy efficient light bulbs, whether he or she shops for food at local stores, and what sort of technology he or she uses for electrical power. Some variables are direct, such as the carbon dioxide released by a person driving a car, while others are indirect and focus on the entire life cycle of products, such as the fuel used to produce the vegetables that a person buys at the grocery store and disposal of packaging waste.

The notion of a carbon footprint is being considered in a wide range of areas, including the construction of low-impact homes, offices, and other buildings. The design must take into account not only the future impact of the building in terms of carbon emissions, but carbon-related production costs for the materials, labor, and energy used to build it. Mathematical modeling and optimization helps engineers and architects create efficient, useful, and sometimes even beautiful structures while reducing the overall carbon footprint. Mathematicians are also involved in the design of technology that is more energy efficient, as well as methods that allow individuals and businesses to convert to electronic documents and transactions rather than using paper. These methods include using improved communication technology, faster computer networks, improved methods for digital file sharing and online collaboration, and security protocols for digital signatures and financial transactions. Manufacturers are increasingly being urged and even required to examine their practices, since manufacturing processes produce both greenhouse gasses from factory smokestacks and waste heat. Mathematicians and scientists are working on ways to recycle much of this heat for power generation. One proposed device combines a loop heat pipe, which is a passive system for moving heat from a source to another system, often over long distances, with a Tesla turbine. Patented by scientist and inventor Nikola Tesla, a Tesla turbine is driven by the boundary layer effect rather than fluid passing over blades as in conventional turbines. It is sometimes called a Prandtl layer turbine after Ludwig Prandtl, a scientist who worked extensively in developing the mathematics of aerodynamics and is credited with identifying the boundary layer.

These are in turn related to the Navier–Stokes equations describing the motion of fluid substances, named for mathematicians Claude-Louis Navier and George Stokes. The Navier–Stokes equations are also of interest to pure mathematics, since many of their mathematical properties remain unproven at the beginning of the twenty-first century.

Carbon Footprints of People

A calculation of the carbon footprints of different aspects of people’s lives, and then the aggregate for a year, is always an estimate. For example, different towns use different methods for generating electricity. Entering data for an electric bill allows for a rough estimate of the household’s carbon footprint, but not exact numbers, which would depend on the electricity generating methods. Houses contribute to carbon footprints through their building costs, heating and cooling, water filtration, repair, and maintenance—all of which use products with carbon footprints.

Travel is another major contributor to peoples’ carbon footprints. Daily commutes and longer trips with any motorized transportation contribute to carbon dioxide emissions. When computing carbon footprints, fuel production and storage costs have to be taken into consideration.

The food that people eat contributes to the carbon footprint if it is transported by motorized vehicles before being eaten. The movement of locavores (people who eat locally grown foods) aims to minimize the carbon footprint of food. Also, different farming practices may contribute more or less to the carbon footprint of food.

The objects people use contribute to their carbon footprints. Recycling and reusing reduces the need for landfills, waste processing, and waste removal, all of which have carbon footprints. There are individuals and communities who avoid waste entirely; several countries, such as Japan, have plans to mandate zero-waste practices within the next few decades.

Economy and Policy

There are two main strategies for addressing carbon footprints. The first strategy is to lower the carbon footprint by modifying individual behaviors, such as traveling by bike, eating locally, and recycling. The second strategy is to perform activities with negative carbon footprints, such as planting trees, to match carbon footprints of other activities.

Some companies incorporate activities that offset the carbon footprint of their main production into their business plans, either lowering their profit margins or passing the cost to their customers. There are economic laws and proposals that attempt to integrate carbon footprint considerations into the economy, usually through taxes on use of fuel, energy, or emissions. Carbon dioxide emissions, in economic terms, are a negative externality (a negative effect on a party not directly involved in the economic transaction). Money collected through carbon taxes is generally used to offset the cost to the environment.

Emissions trading is another mathematics-rich area of dealing with carbon footprints economically. Governments can sell emission permits to the highest-bidding companies, matching their carbon footprints, and capping the total emission permits sold. This method allows prices of permits to fluctuate with demand, in contrast with carbon taxes in which prices are fixed and the quantities of emissions can change. Economists model the resulting behaviors, and advise policymakers based on the models’ outcomes.

Marginal Abatement Cost Curve

“Marginal cost” is an economic term that means the change of cost that happens when one more unit of product is made, or unit of service performed. For physical objects, the curve is often U-shaped. The first units produced are very costly because their cost production involves setting up the necessary infrastructure. As more units are produced, and the infrastructure is reused, the price goes down until the quantities of production reach such levels that the logistic difficulties drive the price per additional units higher again.

A marginal abatement curve shows the cost of reducing emissions by one more unit. These curves are usually graphed in percents. For example, such a curve can be a straight line, with the cost of eliminating the first few percent of emission being zero or even negative. This happens because it can be done by changing practices within existing economic infrastructures, such as cheap smart switches into the residential sector’s lighting grids. Additional lowering of the carbon footprint, however, requires deeper and costlier changes to the way of life. For example, there are relatively high costs involved in switching to wind and solar power, or switching to the use of crop rotations that do not require high-carbon fertilizers.

Country by Country

The average carbon footprint of citizens varies by country. For example, in late 2000s, the average annual carbon footprint of a U.S. citizen was about 30 metric tons per year, and a Japanese citizen about 10 metric tons per year. However, these calculations are extremely complicated because of global trade. For example, many developed countries “export” or “outsource” their carbon emissions to developing countries. Products imported from developing countries account for anywhere from a tenth to a half of the carbon footprints of developed nations.

International calculations indicate a strong correlation between the average carbon footprint of a country’s citizen and the average per capita consumption. The higher the consumption rates, the higher the average carbon footprint.

The categories used for calculation for countries are similar to those used for individuals and include construction, shelter, food, clothing, manufactured products, services, transportation, and trade. The ratios of these items to one another in the carbon footprints vary by country. For example, the greatest item in the U.S. carbon footprint is shelter (25%), with mobility being second (21%). In contrast, Canada’s greatest item affecting carbon footprint is mobility (30%), and its second greatest is shared between shelter and service (18% each).

Bibliography

Berners-Lee, Mike. How Bad Are Bananas? The Carbon Footprint of Everything. Vancouver, BC: Greystone Books, 2011.

Goleman, Daniel. Ecological Intelligence: How Knowing the Hidden Impacts of What We Buy Can Change Everything. New York: Broadway Books, 2009.