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Modeling Treatment And Analysis Of Water
Flow Problem In Water Tower
Abstract
In daily life and industrial production, water storage and consumption management is a
crucial task. Especially in the increasingly tense water resources, the real-time monitoring and
accurate management of water level in the tank becomes particularly important. This study
focuses on a specific scenario, that is, the variation of water level in a cylindrical tank. The
diameter of the tank is known, and depending on the water level, there is a pump that will start or
stop working at a specific amount of water. By collecting and analyzing water level data from this
tank over the course of a day, we can gain a deeper understanding of water consumption patterns
and develop more effective water management strategies accordingly.
The main purpose of this study is to build a function model f(t) of water consumption based
on the given data through interpolation and fitting methods in order to predict and control water
use. Through this model, we can answer a series of key questions, such as the maximum time and
volume of water consumption in the tank, the time point when the water consumption reaches half
of the maximum, and the time period when the maximum water consumption occurs. In addition,
we will pay special attention to the water consumption during the operation of the pump and
calculate the total water consumption throughout the day. In order to simulate the water level
change more accurately, we will use the cubic spline interpolation method to establish the water
level function H(t). This method is widely used in numerical analysis because of its ability to
generate smooth and continuous functions while preserving local features of the data. By showing
the computational process for solving a system of linear equations, we will ensure the accuracy
and reliability of the model.
Finally, considering the possible leakage situation in practical applications, we will further
explore a special scenario: when there is a small leak hole at the bottom of the tank, how the water
leakage rate will affect the water level change. By establishing a differential equation describing
