Sustainable Thermal Energy Storage Systems: A Mathematical Model of the “Waru-Waru” Agricultural Technique Used in Cold Environments

The provision of food in pre-Inca/Inca cultures (1000 BC–≈1532 AD) in environments near Lake Titikaka (approximately 4000 m above sea level) was possible through an agricultural technique called “Waru-Waru”, which consists of filling the space (volume) between rows of land containing plants that are cultivated (a series of earth platforms surrounded by water canals) with water, using water as thermal energy storage to store energy during the day and to regulate the temperature of the soil and crop atmosphere at night. The problem is that these cultures left no evidence in written documents that have been preserved to this day indicating the mathematical models, the physics involved, and the experimental part they performed for the research, development, and innovation of the “Waru-Waru” technique. From a review of the existing literature, there is (1) bibliography that is devoted to descriptive research (about the geometry, dimensions, and shapes of the crop fields (and more based on archaeological remains that have survived to the present day) and (2) studies presenting complex mathematical models with many physical parameters measured only with recently developed instrumentation. The research objectives of this paper are as follows: (1) develop a mathematical model that uses finite differences in fluid mechanics, thermodynamics, and heat transfer to explain the experimental and theory principles of this pre-Inca/Inca technique; (2) the proposed mathematical model must be in accordance with the mathematical calculation tools available in pre-Inca/Inca cultures (yupana and quipu), which are mainly based on arithmetic operations such as addition, subtraction, and multiplication; (3) develop a mathematical model in a sequence of steps aimed at determining the best geometric form for thermal energy storage and plant cultivation and that has a simple design (easy to transmit between farmers); (4) consider the assumptions necessary for the development of the mathematical model from the point of view of research on the geometry of earth platforms and water channels and their implantation in each cultivation area; (5) transmit knowledge of the construction and maintenance of “Waru-Waru” agricultural technology to farmers who have cultivated these fields since pre-Hispanic times. The main conclusion is that, in the mathematical model developed, algebraic mathematical expressions based on addition and multiplication are obtained to predict and explain the evolution of soil and water temperatures in a specific crop field using crop field characterization parameters for which their values are experimentally determined in the crop area where a “Waru-Waru” is to be built. Therefore, the storage of thermal energy in water allows crops to survive nights with low temperatures, and indirectly, it allows the interpretation that the Inca culture possessed knowledge of mathematics (addition, subtraction, multiplication, finite differences, approximation methods, and the like), physics (fluids, thermodynamics, and heat transfer), and experimentation, with priority given to agricultural techniques (and in general, as observed in all archaeological evidence) that are in-depth, exact, practical, lasting, and easy to transmit. Understanding this sustainable energy storage technique can be useful in the current circumstances of global warming and climate change within the same growing areas and/or in similar climatic and environmental scenarios. This technique can help in reducing the use of fossil or traditional fuels and infrastructure (greenhouses) that generate heat, expanding the agricultural frontier.