Conical Tank Water Drainage: Finding Time to Empty with Differential Equations

In summary, the problem involves an inverted conical tank with water draining at a rate proportional to the depth of water. The differential equation for this scenario is dy/dt = -k(y0/r0)^2 (1/piy). To find the time it will take for the tank to empty, the differential equation can be used to obtain y as a function of time, which can then be solved to find the time when y=0.
  • #1
chae
1
0

Homework Statement


Water drains out of an inverted conical tank at a rate proportional to the depth (y) of water in the tank. Write a diff EQ as a function of time.

This tank's water level has dropped from 16 feet deep to 9 feet deep in one hour. How long will it take before the tank is empty.

Homework Equations



V=1/3pir2y

dV/dt=pi(r0/y0)2y2dy/dt

The Attempt at a Solution


The solution to the first question is: dy/dt=-k(y0/r0)2(1/piy)

I don't really know how to go about finding the time it will take to empty the tank.

Please help!
 
Last edited:
Physics news on Phys.org
  • #2
hey, welcome to physicsforums! your solution to the first question is correct. (I'm guessing you're using 'k' as a constant. this is good.) So now, you need to make use of this differential equation to get 'y' as a function of time. You have hopefully done this kind of integral before. It just needs a bit of rearranging to get a nice answer.
 

FAQ: Conical Tank Water Drainage: Finding Time to Empty with Differential Equations

What is a draining conical tank?

A draining conical tank is a type of tank that has a conical or cone-shaped bottom, which allows for the efficient draining of liquids or fluids.

How does a draining conical tank work?

A draining conical tank works by utilizing the force of gravity to drain liquids or fluids from the bottom of the tank. The conical shape of the tank allows for a more efficient and faster draining process.

What are the advantages of using a draining conical tank?

The advantages of using a draining conical tank include faster and more efficient draining of liquids, as well as the ability to completely drain the tank without any residual liquid remaining. The conical shape also allows for better mixing of the fluid inside the tank.

How do you calculate the draining time for a conical tank?

The draining time for a conical tank can be calculated using the formula t = (A*H) / (2G), where t is the draining time, A is the cross-sectional area of the base of the tank, H is the height of the tank, and G is the acceleration due to gravity. This formula assumes that the draining is occurring through a small hole at the bottom of the tank.

What are some common applications of draining conical tanks?

Draining conical tanks are commonly used in industries such as food and beverage processing, chemical and pharmaceutical production, and wastewater treatment. They are also used in everyday household appliances such as washing machines and dishwashers.

Similar threads

Time to Empty a Tank with Water Flowing at 2h kg/s
Replies
1
Views
1K
When is the tank empty? [differential equation]
Replies
3
Views
2K
Water drains from a tank. Write a differential equation.
Replies
1
Views
4K
Conical Tank Water Leak Rate Calculation
Replies
2
Views
3K
ODE problem -- Find the amount salt in the tank as water flows through it
Replies
3
Views
3K
How Does Water Depth Change in a Conical Tank?
Replies
2
Views
2K
Finding the Rate of Water Input in a Leaking Conical Tank
Replies
1
Views
1K
Ordinary Differential Equation - tank with inflow and outflo
Replies
1
Views
2K
Work problem -- lifting water out of tanks
Replies
2
Views
2K
Empty Conical Tank: Use Torricelli's Principle to Find Time
Replies
1
Views
4K

Hot Threads

Recent Insights

Back
Top