Solving Drain Problem: Salt Concentration in Effluent

[walisyh]

drain problem?

Homework Statement

A tank has 1000 m3 of salt solution. The salt concentration is 10 kg/m3. At time zero, salt-free water starts to flow into the tank at a rate of 10 m3/min. Simultaneously salt solution flows out of the tank at 10 m3/min, so that the volume of the solution in the tank is always 1000 m3. A mixer in the tank keeps the concentration of of salt in the entire tank constant; the concentration in the effluent is the same at the concentration in the tank. What is the concentration in the effluent as a function of time?

 

[tiny-tim]

welcome to pf!

hi walisyh! welcome to pf!

show us what you've tried, and where you're stuck, and then we'll know how to help!

 

[walisyh]

hi walisyh! welcome to pf!

show us what you've tried, and where you're stuck, and then we'll know how to help!

ok what i have done is that:
when 10m3/min fresh water entered into tank 10 m3/min of mixture leaves the tank
now in 1 minute 100 kg of salt is being removed from the tank so conc. of effluent is 10kg of salt/m3 in 1 minute, in 2 min it 9.9 kg/m3 of salt and decreases approx 0.1 after every minute. I am stuck in making the general expression of effluent conc. as fuction of time.
what i have done is that y(t)=10-(t-1)/10

 

[tiny-tim]

hi walisyh!

you're looking for a differential equation …

if the weight of salt is W(t) at time t, what is dW/dt ?

(or if you prefer, you can use the concentration C(t) instead of the weight)

 

[Robert1986]

I think that this problem warms the cockles of the heart of every math student / mathematician. This is one of those problems that every single ODE student has done. I love it! But, once you have the equation written, you'll probably see how to do it.

 

[walisyh]

I think that this problem warms the cockles of the heart of every math student / mathematician. This is one of those problems that every single ODE student has done. I love it! But, once you have the equation written, you'll probably see how to do it.

can you give me the equation? Because this question was given by my fluid teacher

 

[jhae2.718]

Here are some thoughts:

• You are interested in the concentration as a function of time.
• The volume of the tank is constant.
• Salt water is leaving the tank at -10 m³/min
• The salt water and fresh water form a homogenous solution.
• How can you define concentration?
• From this, what happens if you differentiate this expression?
• Can you write the rate at which the mass of salt water is changing as a function of time?

(This is a rather fun and simple ODE...)

 

[walisyh]

finally i have done
Every minute we remove 1% of the salt. So the concentration at t minutes is (0.99^t)*(10 kg/m^3).

if anyone can give another solution pls post

 

[tiny-tim]

hi walisyh!

(try using the X² icon just above the Reply box )

Every minute we remove 1% of the salt. So the concentration at t minutes is (0.99^t)*(10 kg/m^3).

that would be correct if the inflow was shut off for a minute while the outflow continued, then the tank was refilled, then the inflow was shut off for another minute, and so on

but the inflow is continuously altering the concentration, so you need a differential equation …

try again ​

 

[walisyh]

another attempt
Suppose the concentration at time t is c(t) so c(0)=10kg/m³ .

You have c ′ (t)=(−10m ³ /min) /1000m ³ c(t)
so c(t)=ke^(−t/100) for some constant k and from the starting condition k=10kg/m ³ so

c(t)=10e^{(−t /100 )}kg/m ³.

 

[tiny-tim]

Woohoo! ​