Solving Poisson Equation to Pump Water from Drains

In summary, the conversation is about finding a numerical solution to the Poisson equation in order to calculate the amount of water pumped from a drainage system. The suggested approach is to discretize the region and use iteration to solve the equation at each point. A recommended resource for solving this problem is the book "Numerical Recipes" by William H. Press et al.
  • #1
darklide
13
0

Homework Statement


numerical solution of poisson equation with application to the pumping of water from drains


1. time
2. Amount of water has been pumped

Homework Equations


The poisson equation


The Attempt at a Solution


1. I know i have to use iteration
2. I am still searching notes on the web concerning that

I just got that yesterday and i have little time to do it. I'll post attempts as i get them. But if anyone can tell me where to find about the question that would greatly help. Then i'll just have to implement that using excel which will be a piece of cake(i hope lol)
 
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  • #2


Hello,

Thank you for your post. It sounds like you have a challenging problem to solve. The Poisson equation is a mathematical equation that describes the behavior of electric potential in a given region. It can also be applied to other physical phenomena, such as the flow of water in a drainage system.

To solve this problem, you will need to use numerical methods, such as iteration, to approximate the solution. One approach you could take is to discretize the region into a grid and solve the equation at each point on the grid. This will give you a numerical solution for the electric potential at each point, which can then be used to calculate the flow of water.

There are many resources available online for solving the Poisson equation numerically. One helpful resource is the book "Numerical Recipes" by William H. Press et al. This book provides a comprehensive overview of various numerical methods and includes examples and code for solving the Poisson equation.

I hope this helps you get started on your problem. Good luck!
 
  • #3


I would first commend the student for recognizing the importance of numerical solutions in solving real-world problems such as pumping water from drains. The use of the poisson equation is a valid approach, as it can accurately model the behavior of fluids in a confined space.

To solve this problem, the student is correct in stating that iteration is necessary. This involves breaking down the problem into smaller steps and repeating them until a desired result is achieved. In this case, the steps would involve using the poisson equation to calculate the pressure distribution in the drain, and then using that information to determine the amount of water that can be pumped out over a given time period.

To find resources on solving the poisson equation numerically and its application to pumping water from drains, the student can refer to textbooks and research articles on numerical methods in fluid mechanics. Additionally, there are many online resources and tutorials available that can guide the student through the process of solving the poisson equation and applying it to this specific problem.

Finally, as a scientist, I would encourage the student to not only focus on finding a solution to the problem, but also to understand the underlying principles and assumptions involved. This will not only help in finding an accurate solution, but also in applying it to other similar problems in the future.
 

1. How does solving the Poisson equation help in pumping water from drains?

The Poisson equation is a mathematical formula that helps us model and understand the flow of fluids, such as water. By solving this equation, we can determine the relationship between the pressure and velocity of water in a drain, and use this information to design efficient pumping systems.

2. What is the Poisson equation and how is it derived?

The Poisson equation is a partial differential equation that describes the relationship between the pressure and velocity of a fluid. It is derived from the Navier-Stokes equations, which govern the motion of fluid particles, and the continuity equation, which ensures that mass is conserved.

3. What factors affect the solution of the Poisson equation for pumping water from drains?

The solution of the Poisson equation is affected by several factors, including the geometry of the drain, the properties of the fluid, and the boundary conditions (such as the pressure at the drain entrance and the velocity at the drain exit). These factors can be adjusted to optimize the pumping process and ensure efficient removal of water from drains.

4. Can the Poisson equation be solved analytically or does it require numerical methods?

The Poisson equation can be solved analytically for simple geometries and boundary conditions. However, for more complex systems, numerical methods are typically used to obtain an accurate solution. These methods involve breaking the problem into smaller, solvable parts and using algorithms to find an approximate solution.

5. How is the solution of the Poisson equation used in practical applications for pumping water from drains?

The solution of the Poisson equation is used in practical applications by engineers and designers to determine the optimal size and placement of pumps, as well as the required pressure and flow rate to efficiently remove water from drains. This helps to minimize energy consumption and maximize the effectiveness of drainage systems.

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