Solution pumped into a tank problem.

[derail]

Homework Statement

Suppose that a large mixing tank initially holds 400 gallons of water in which 65 pounds of salt have been dissolved. Another brine solution is pumped into the tank at a rate of 6 gal/min, and when the solution is well stirred, it is pumped out at a faster rate of 8 gal/min. If the concentration of the solution entering is 3 lb/gal, determine a differential equation for the amount A(s) of salt in the tank at time t.

Homework Equations

Flow rate is Volume times cross sectional area, however, I do not know if that is relevant, it has been a while since I have done something like this.

 

[mfb]

Flow rate is Volume times cross sectional area, however, I do not know if that is relevant, it has been a while since I have done something like this.

Geometry does not matter here.

If the concentration of salt is x and 8 gallons/min flow out, can you determine the rate of salt flowing out?
If 6gallons/min with a salt concentration of 3pounds/gallon (imperial units are weird) flows in, can you determine the rate of salt flowing in?

This should help to get your differential equation.

 

[derail]

So wouldn't the 8 gallons/min be the flow rate?

I would think i can find the rate of salt flowing out by taking the 6 gallons/min and the 3 pounds/gallon and trying to get pounds/minute by multiplying them to get 18 pounds/minute.

 

[haruspex]

So wouldn't the 8 gallons/min be the flow rate?

There are two flow rates, one in, one out.

I would think i can find the rate of salt flowing out by taking the 6 gallons/min and the 3 pounds/gallon and trying to get pounds/minute by multiplying them to get 18 pounds/minute.

You mean the rate of salt flowing in, right? You know the flow rate of water out, so you need the concentration of salt in that. You have the variable A(t) for the amount of salt in the tank at time t, so you need to know how much water that's dissolved in. Can you find an equation for the volume of water in the tank at time t?

 

[derail]

I am not sure how to incorporate the flows of the liquid into the equation when I am ultimately looking for an equation for the amount of salt.

 

[haruspex]

You know the flow rates of water in and out, so how much water is in the tank at time t?

 

[derail]

I see, we should be able to use the same logic to make is so that the units of the A(t) are lbs per minute by using the concentration of salt water. Do you have a suggestion on how to write the equation?

I figure the way we are doing this, we are basically adding two separate equations, one for the flow rate going in and one for the rate going out.

 

[haruspex]

Forget the salt for the moment. There's 6g/m flowing in, 8g/m flowing out. What's the rate of change of volume of water? The tank started with 400 gallons. How many after t minutes?

 

[derail]

So the rate of change is 2g/m flowing out. The tank started with 400 gallons.

 

[derail]

So after t mins we should have, 2*t+400 gallons of water.

 

[haruspex]

So after t mins we should have, 2*t+400 gallons of water.

Is the volume of water increasing or decreasing?

 

[derail]

Decreasing my bad, so we actually get -2*t+400, now to write that equation.

 

[derail]

So the water flow rate =(-2*t)+400, and we know that 3 pounds/gallon is being pumped in so we should be able to write: =((-2*t)+400)*3 to correct the units for Pounds per minute?

 

[haruspex]

So the water flow rate =(-2*t)+400,

No, that's not a flow rate at all. That's the volume of water in the tank at time t.
If the salt in the tank at time t is A(t), what's the concentration in the tank at time t?

 

[derail]

I am trying to find a differential equation for the amount A(s), meaning the amount of salt. I miss wrote that last equation, you are right. However, I am trying to get the amount of salt, so I figure if I know the amount of water then I can use the concentration to get the amount of salt? like I said in reply #13

If the salt in the tank at time t is A(t), what's the concentration in the tank at time t?

Well the concentration of the solution entering is 3lb/g.

 

[derail]

So my thought process was, we can take the flow rate equation and incorporate the concentration of salt to water?

 

[haruspex]

I am trying to find a differential equation for the amount A(s),

Yes, I know, and I'm trying to lead you to that by asking a series of very simple questions. So please, try to answer them, not a question I didn't ask.
You have calculated the volume of water in the tank at time t. The amount of salt in the tank at time t is A. (Which one writes as A=A(t), or if you want to call the variable A_s then A_s=A_s(t). I don't know why you keep writing A(s). There is no variable s.) So, very simple question, what is the concentration of salt in the tank at time t?