Mathematical Modelling of Non-Stop Heron’s Fountain and its Practical Applications

Abstract

Heron’s fountain is one of the brilliant ideas given by Heron (Mathematician and Physicist) in which we get free kinetic energy to run our fountain. The schematic diagrams of the physical demonstration of the magic fountain are given in this paper. The fundamental work principles of magic fountain are hydraulics and Pneumatics. In engineering, it is important to analyze a system by developing mathematical models, so we used the principle of fluid mechanics and Bernoulli’s theorem to develop the equation for the velocity of a free jet. It is a very important thing in engineering to make use of energy, The free jet having kinetic can used for energy production purposes. In this paper, Mathematical Modelling and some fundamental applications of the non-stop fountain have been discussed. Heron’s fountain due to its perpetual nature has thrilling applications in the micro hydropower sector. Moreover, it has keen value in environmental sustainability and air pollution. The contribution of the fountain cannot be underestimated towards the beauty of buildings and relaxation of mind.

Introduction

I. INTRODUCTION

The Non-Stop fountain concept was first given by a Mathematician and Physicist named Heron of Alexandria in the first century AD. Heron’s Fountain is an amazing mechanical device involving the principal of hydraulics and pneumatics. The basic schematic diagram is shown in Fig. 1. Mathematics is the beauty of science and if we observe the non-stop fountain with an engineering eye it needs mathematical modeling.

In this paper, we briefly derived the equations for pressure differences, Velocity of the resulting out jet. With the help of these equations, we can critically analyze the effect of one constraint on another one. Basically, the main principle in the heron’s fountain is the principle of conservation of energy and the gravitational potential energy is converted into fluid kinetic energy. This paper will show the relationship between a jet’s speed and the potential energy of the water tank.

The mathematical analysis contains some assumptions also we will discuss them later. As we discussed earlier about energy conservation here’s for the derivations of Heron’s fountain equation, we will use Bernoulli’s theorem of energy conservation in two dimensions.

In this Heron phenomenon, the free jet’s kinetic energy is like a free energy source that we can use for many practical purposes. Due to this kinetic energy non-stop, the fountain has many uses and full applications some of which are under discussion like Air quality improvement, Reduction of noise pollution, Cooling, Hydropower generation, and water feature in buildings.

There are some limitations of the non-stop fountain like in Fig. 1 when the middle tank becomes empty the flow will stop.  There should be no loss of air or water because both are working fluids there. We will discuss them and suggest solutions for them.

II. FUNDAMENTAL WORKING PRINCIPLE

The fundamental principles of magic fountains are basically two, the first one is the principal of pneumatic and the second one is the principal of hydraulics.                               

In pneumatics, we make use of compressed gas or air to transfer mechanical energy from one point to another. In Fig. 1. The pressure at point 1 is downward and pushes the water level in tank A upward and as result the air in free space in tank A and tank B first become compressed (because as the water level moves up in tank A it exerts pressure on air).

The hydraulics principle is that this compressed air exerts pressure on the water in tank B and pushes the water downward and as a result, the water moves out to the jet through the pipe.

V. ACKNOWLEDGMENTS

The authors would like to acknowledge all the friends and Professors who motivated and support our research to work on the magic fountain. We specially want to acknowledge our family members who supported us financially for this research.

VI. CONFLICT OF INTEREST

The authors declare that they have no competing interests.

VII. AUTHORS CONTRIBUTION

Mainly the paper is comprised of two portions Mathematical modelling and explanation and Practical applications of the magic fountain. S. Ali worked on understanding the history and working principal of Heron’s fountain. Moreover, he worked on the developing the mathematical equations and applying Bernoulli’s theorem. S.M. Zain along with M. Tariq Shah contributed his efforts towards the practical applications and implementation of the fountain. His work also includes the Graphs, Diagrams, Chats and Tables.

Conclusion

In this article, a critical analysis of a model of Heron’s Fountain was done. Mainly focused on the working principle, mathematical modeling, and practical applications of it. The equation for free jet velocity has been derived and its dependence on different parameters has been highlighted. Working principle circulates about two major principles one is the pneumatics and the other is the Bernoulli’s equation. Results show that the kinetic energy of the free jet is directly proportional to the under root of the distance between tank A and tank C. This means that the greater the distance the greater the velocity. Our discussed model of Heron’s fountain has many applications because of its non-stop water circulation and a jet with sufficient kinetic energy. Some important applications have been discussed and the engineering behind how to apply Heron’s principle for that application.

References

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Copyright

Copyright © 2024 S. Ali, S. M. Zain, M. Tariq Shah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.