Optimizing water distribution

Summary: Mathematicians have long studied issues related to optimizing water distribution.

Water distribution has two separate but interrelated meanings: the natural physical distribution of water in the world and the way in which people choose to distribute available water. In some regions, accessing and distributing fresh water for human needs, like drinking and irrigation, can be a significant challenge. Roughly 70% of the Earth’s surface is covered with water but most is saline (salty). Much of Earth’s fresh water is in glaciers or underground. Some is polluted from human activities. In the early twenty-first century, approximately 20% of Earth’s population lived in areas with insufficient fresh water because of climate or geography. About the same number lived in areas in which water existed but where technological or economic barriers limited effective distribution. Many systems have been devised throughout history and in different societies to access and distribute water. It is so valuable a resource that armed conflicts been fought over water. Mathematicians, scientists, and others who work on water distribution problems use mathematical techniques to design, build, optimize, and monitor water distribution and associated wastewater systems. For example, graph theory is used to model water distribution networks. Graph edges may represent pipes and nodes represent intersections, junctions, and access points. Statistical and topological methods can be used to compare networks in terms of capacity and reliability against failure.

Irrigation

Irrigation is an ancient practice that allows food to be grown where it might otherwise not thrive. Evidence shows that it was used as early as the sixth millennium b.c.e. in Mesopotamia, Egypt, and Persia, and the fifth millennium b.c.e. in South America. In the early twenty-first century, agriculture is still globally the greatest consumer of fresh water, though it varies widely by location. For example, the United Kingdom’s abundant rainfall means that it requires almost no irrigation. Mexico and India, on the other hand, use it extensively. The green revolution of the twentieth century, which greatly increased the agricultural yield of many developing countries, relied in part on irrigation. One criticism was that the increased food production in these areas resulted in accelerated population growth that placed further burdens on scarce water resources. This criticism is supported by some statistics and mathematical models, which show that the demand for water grew at rates that exceeded population increases, raising per-capita water requirements.

Mathematicians and others who study ancient systems of irrigation in order to better understand them (and perhaps improve modern methods) have noted that some societies appear to have created and implemented complex and efficient water distribution methods without using mathematical methods for planning. Others have sought to build mathematical models of irrigation systems. The paddy field system used for growing rice generally requires the creation of intricate structures of terraces, canals, and reservoirs in order to ensure that all fields receive adequate water. It is believed to have been used as early as 4000–3500 b.c.e. in China and Korea. Researchers who have investigated mathematical models to describe a paddy field system have noted that it may not be possible to create a reliable model by including only variables based on physical measures such as amount of water available and rate of evaporation. A variable describing an ethic of cooperation among owners of the various fields, a factor that is difficult to quantify, was also required to ensure that water would be used fairly. For example, if owners on the upstream end of a water source took more than their fair shares, the owners farther downstream would not receive sufficient water for their crop, regardless of the values of some other variables.

Industry

Industry is the second largest category of global water use. Most industrial processes need water in some way, though some are more readily visible, such as hydropower generation of electricity and water extraction of minerals in mining. At the start of the twenty-first century, per capita water use is typically higher in industrialized nations than in developing countries, though this gap is closing. Some economists use the term “virtual water” to refer to the water that is used in the entire chain of manufacturing a product or growing an agriculture commodity. Similar to a carbon footprint, which is often used to quantify the quantity of greenhouse gasses emitted by a process, a water footprint represents the total amount of water used to create a good or service. Calculating water footprints provides an additional metric for assessing and comparing the environmental impact of competing products and services. For example, in 2010, the Water Footprint Network estimated that production of 1 kilogram of beef required about 16,000 liters of water, while one kilogram of rice required 3000 liters of water, and one liter of milk required 1000 liters of water.

Sanitation

The creation of sanitary systems of water supply and wastewater disposal or treatment is a major factor in the general improvement of public health from about the mid-nineteenth century onwards. Large cities, such as London, New York, and Boston, were among the first to establish municipal water supply systems. They were motivated in part by data collected by statisticians and others such as physician John Snow, who demonstrated via statistical methods that an 1854 cholera outbreak in London could be traced to the local water pump.

Mathematical methods may be used to model different aspects of supply systems. The fluid pressure necessary for water to flow through a system is affected by variables like gravity. Water stored in a rooftop tank will deliver water at a higher pressure to lower floors versus higher ones. Mathematical calculations show that a vertical foot of water exerts a pressure of 0.433 pounds per square inch (psi) at its bottom surface. The flow of water through the system is a function of the cross-sectional area of the pipe: Q=A×V, where Q is the flow of water through the system, A is the cross-sectional area of the pipe, and V is the velocity of the water.

Municipal water systems tend to be quite complex, involving massive networks of storage tanks, pipes, pumps, and valves. Mathematical models are used to describe and manage these systems. Navier–Stokes equations, named for mathematicians Claude-Louis Navier and George Gabriel Stokes, are partial differential equations that describe fluid flow and velocity, while the Reynolds number, named for mathematician Osborne Reynolds, quantifies “laminar” (smooth) and turbulent fluid flow through a pipe. Contamination is an ever-present risk because of the natural physical deterioration of system components over time (such as corroded pipes) as well as the possibility of accidental or deliberate introduction of contaminants. Researchers are developing systems that can sense when a contaminant has been introduced into the water distribution system, allowing for rapid identification of the time and location of its introduction. For example, experiments done by the U.S. Environmental Protection Agency showed statistically that chlorine and total organic carbon, which are routinely monitored in municipal water systems, were sensitive and reliable predictors of contamination.

Bibliography

Cohen, Y. Koby. Problems in Water Distribution: Solved, Explained, and Applied. Boca Raton, FL: CRC Press, 2002.

Gates, James. Applied Math for Water Distribution, Treatment, and Wastewater Operators. Dubuque, IA: Kendall Hunt Publishing, 2010.