Optimal Shape of Water Towers

Abstract

Abstract. Water tanks can be seen in communities all across the country, but how well are they designed? In this work, we study the optimal shape of water tanks to find the fastest emptying time. In particular, we compare emptying the efficiency for different shaped water towers with the same volume, height, and cross-sectional outlet area. We first review a formula for the emptying time as a function of the volume and the tank's height and then compute the emptying time for several specific tank shapes. The question of whether there exists a tank with a minimal emptying time is also considered in the context of prior work. Finally, our added contribution is to fix the volume of a typical water tank and compute the area and the emptying time for different tank shapes, including composites, in order to develop an understanding for how an optimal tank might be designed.Key words. Water towers, Optimal shapes, Emptying efficiency 1. Introduction. Since the dawn of civilization, the retrieval of water has been a fundamental problem for humanity. To address this problem, early humans would be forced to settle near a body of water. Water was primarily used for drinking, sanitation, and watering crops [10]. However, as cities were built and infrastructure became more common, the need for water in each individual building became more apparent along with the rise of the amount of uses of water. Thus, the need to efficiently move water from one place to another quickly arose. One such solution came in the form of an aqueduct in the Roman times. However, the Minoans, who were much before their time in terms of water systems, used aqueducts before the Romans during the Bronze age [2]. The Minoans had aqueducts, a filtration system, harvested rain water, sewage pipes, etc. Specfically, the aqueduct made it possible for humans to live farther away from bodies of water; however, this range remained limited. The logistics of water remains an issue due to the necessity of water for human life. Storing water for immediate use at specific times has become the modern day equivalent of these logistical issues. The most common way to store water is to use a water tower [2].Water towers are basic water storage devices capable of holding hundreds of thousands to millions of gallons of water at a time. We can think of water towers as simply water tanks that are elevated off the ground. This is done to allow gravity to enhance water pressure in order to get the water where it needs to go [7]. For each foot of the water tower, the tower increases its water pressure by roughly 0.43 pounds per square inch. An essential feature in the present day is the ability of a water tower to hold a day's worth of water in case of emergency. For example during a power outage, the mechanical pumps cannot work and the people must rely on the backup water supply the water tower provides. Water towers are also used during parts of the day when water demand is high to alleviate strain on the system. Usually the busiest time is in the mornings, when people ar...