Continuously draining tank (constant input\dependant output)

In summary: V and h is just V = Ah. Another assumption I'm making is that the bathtub has a constant level of water (ie. no inflow or outflow during the time frame I'm considering).
  • #1
Mike T
2
0

Homework Statement


I've been asked to solve a problem where a bathtub is being filled at a constant rate, meanwhile water is draining out continuously. The bathtub fills with a constant flow rate of [itex] \frac{dI}{dt} [/itex] and drains with a flow rate [itex] \frac{dO}{dt} [/itex] which is directly proportional to √h where h is the height of the water.

The question is to come up with an equation describing the height of the water over time, given this constant input and varied output. Use this to find out what time the bath water will overflow.

Homework Equations



[itex] netflow = \frac{dI}{dt} - \frac{dO}{dt} [/itex]

3. The Attempt at a Solution

So I've assumed that I should come up with a an equation in the form [itex]h(t)[/itex] where h is the height, and t is the time variable.

I know the volume of the bath, so what height needs to be reached. I've tried integrating both of the flow rates, and then combining them e.g. [itex]h = I(x) - O(x)[/itex], and then solving for the initial condition h = bath height. This gives me the time to fill the bath (although I don't now how accurate it is!) - how I can I make this into a more general solution to find the height at any time t?

I'm having great difficulty relating the input and output in this equation together, and would welcome any points on the general approach, as well as my current approach above.
 
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  • #2
You need to write down a differential equation. Before you have done that you cannot integrate it properly. How do you relate the net flow to the rate of change in height?
 
  • #3
You are given that the bath tub drain at a rate "directly proportional to [itex]\sqrt{h}[/itex]". How do you write that as an equation?
 
  • #4
Mike T said:

Homework Statement


I've been asked to solve a problem where a bathtub is being filled at a constant rate, meanwhile water is draining out continuously. The bathtub fills with a constant flow rate of [itex] \frac{dI}{dt} [/itex] and drains with a flow rate [itex] \frac{dO}{dt} [/itex] which is directly proportional to √h where h is the height of the water.

The question is to come up with an equation describing the height of the water over time, given this constant input and varied output. Use this to find out what time the bath water will overflow.

Homework Equations



[itex] netflow = \frac{dI}{dt} - \frac{dO}{dt} [/itex]

3. The Attempt at a Solution

So I've assumed that I should come up with a an equation in the form [itex]h(t)[/itex] where h is the height, and t is the time variable.

I know the volume of the bath, so what height needs to be reached. I've tried integrating both of the flow rates, and then combining them e.g. [itex]h = I(x) - O(x)[/itex], and then solving for the initial condition h = bath height. This gives me the time to fill the bath (although I don't now how accurate it is!) - how I can I make this into a more general solution to find the height at any time t?

I'm having great difficulty relating the input and output in this equation together, and would welcome any points on the general approach, as well as my current approach above.

What is the relationship between the volume of water (V) and the height of water (h)? If it is a cylindrical bathtub (with vertical sides) the relationship is just V = Ah, where A = area of base. However, for a bathtub with sloping sides and rounded bottom, etc., the relationship could be a lot more complicated. So, you had better start by stating an assumption about the bathtub's shape.

Then, if r = volumetric inflow rate (which you called dI/dt), what is the volume of inflow in the time interval ##(t,t + \Delta t)##, and what is the volume of outflow during this same interval? What is the NET inflow volume in that interval? (Here, ##\Delta t > 0## is a very small time increment.)
 
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  • #5
Ray Vickson said:
What is the relationship between the volume of water (V) and the height of water (h)? If it is a cylindrical bathtub (with vertical sides) the relationship is just V = Ah, where A = area of base. However, for a bathtub with sloping sides and rounded bottom, etc., the relationship could be a lot more complicated. So, you had better start by stating an assumption about the bathtub's shape.

Then, if r = volumetric inflow rate (which you called dI/dt), what is the volume of inflow in the time interval (t,t+Δt)(t,t + \Delta t), and what is the volume of outflow during this same interval? What is the NET inflow volume in that interval? (Here, Δt>0\Delta t > 0 is a very small time increment.)

Hey, thanks Ray - in this case, I'm assuming a bath tub has vertical edges, so the volume is directly proportional to the height helpfully. So in this time period [itex]\triangle t [/itex] the net flow in is [itex]r \triangle t [/itex].

Calculating the net outflow in this time period is what I'm finding hard as it is dependant on the volume (or height) of the water. Ah, is this where I use the net inflow volume, [itex]r \triangle t[/itex] as the basis for calculating the outflow volume in this time? This makes sense (if I've got it correct!) for a small period, [itex]\triangle t[/itex], but it wouldn't for a large period. So I need to find the definite integral between v amd max volume?

edit: I mean, integrate from t=0, to t=infinity of course, and then find at what value of t, the volume matches my expected result?
 
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Related to Continuously draining tank (constant input\dependant output)

What is a continuously draining tank?

A continuously draining tank is a system in which fluid is constantly being input into the tank while fluid is also being drained from the tank at a dependent rate. This results in a constant fluid level within the tank.

How does a continuously draining tank work?

A continuously draining tank works by having an input valve or pump that allows fluid to enter the tank, and an output valve or pump that controls the rate at which fluid is drained from the tank. The input and output rates are dependent on each other, resulting in a constant fluid level within the tank.

What are the applications of a continuously draining tank?

A continuously draining tank has various applications, including in chemical and industrial processes, water treatment systems, and irrigation systems. It can also be used in experimental setups to model real-world systems with constant input and dependent output.

What factors can affect the performance of a continuously draining tank?

The performance of a continuously draining tank can be affected by factors such as the input flow rate, output flow rate, and the size and shape of the tank. Changes in these factors can result in fluctuations in the fluid level within the tank.

Are there any limitations to using a continuously draining tank?

One limitation of a continuously draining tank is that it can only maintain a constant fluid level if the input and output rates remain dependent on each other. If there are any changes in these rates, the fluid level within the tank will be affected. Additionally, the design and setup of the tank must be carefully considered to ensure proper functioning and accurate representation of real-world systems.

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