Fluid Dynamics Sand and Water

[aquaventions]

Homework Statement

A tank at sea level with fresh water measuring 36"x19"x30" contains a one inch bed of sand. The sand particles are .21 mm in size and have a specific density of 2.6.

The water in the tank recirculates at a rate of 1,000 gph. There is a one inch return at the surface of the tank, and there are three two inch drains drilled into the base of the tank.

The question is, what is the minimum height of the drain above the surface of the sand such that the sand will not be sucked down the drains?

Homework Equations

The water flow rate provides the force. The particle size and specific gravity provide the mass. We want acceleration of the sand particles to remain at zero.

The equation should be some variation of F=ma, but the actual equation is not provided.

The Attempt at a Solution

This is an actual situation I face with an aquarium that I can neither solve myself nor obtain assistance from anyone I know. I place the question here, because it most closely approximates a home work question.

I am looking for assistance in obtaining possible solutions or at least an equation that expresses the problem.

 

[KataKoniK]

Thank you for your question. This is an interesting problem that requires a combination of fluid mechanics and Newton's laws of motion. The first step in solving this problem is to understand the forces at play.

The force that is causing the sand particles to potentially be sucked down the drains is the force of the water flow. This force is due to the pressure difference between the surface of the water and the drains at the base of the tank. This pressure difference is caused by the flow rate of the water. The equation that relates these variables is known as Bernoulli's equation:

P1 + 1/2ρv1^2 + ρgh1 = P2 + 1/2ρv2^2 + ρgh2

Where P is pressure, ρ is density, v is velocity, and h is height.

In this case, we are interested in the pressure difference between the water surface and the drains, so we can simplify the equation to:

P1 + ρgh1 = P2 + ρgh2

The force of the water flow can be calculated using the equation F=ρAv, where ρ is the density of water, A is the cross-sectional area of the drain, and v is the velocity of the water flow.

Now, we need to consider the forces acting on the sand particles. The force of gravity is pulling the particles down, while the force of the water flow is pushing them up. In order for the sand particles to remain at rest, these two forces must be equal and opposite. This can be expressed as:

Fgravity = Fwater flow

mg = ρAv

Where m is the mass of the sand particles, g is the acceleration due to gravity, and A and v are the same as before.

Now, we can substitute in the values given in the problem to solve for the minimum height of the drain above the sand surface. I will leave this part to you as an exercise, but here is the final equation:

h2 = (P1-P2)/(ρg) - (m/ρA)

I hope this helps you in finding a solution to your problem. Please let me know if you have any further questions or need clarification on any of the concepts. Best of luck!

 

[Mene]

I would first start by identifying the key variables in this problem: tank dimensions, sand particle size and density, water flow rate, and drain height. Then, I would consider the forces at play, such as gravity, buoyancy, and water flow. From there, I would use the equation F=ma to calculate the force on the sand particles and determine the minimum height of the drain that would prevent the sand from being sucked down.

However, without knowing the specific equation or having more information about the system, it is difficult to provide a precise solution. I would suggest consulting a fluid dynamics expert or conducting further research to gather more data and refine the problem. Additionally, considering the practical limitations and potential complications of a real world system, it may be necessary to test and adjust the solution through trial and error.