ODE problem -- Find the amount salt in the tank as water flows through it

In summary, the problem involves a tank initially containing 100 gallons of fresh water and then being filled with salt water at a rate of 2 gallons per minute. After 10 minutes, fresh water is added at the same rate while the mixture continues to leave. The goal is to find the amount of salt in the tank after an additional 10 minutes. The solution involves using the general form of the solution for the amounts of salt in the tank at different times.
  • #1
yecko
Gold Member
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15

Homework Statement



A tank originally contains 100 gal of fresh water. Then water containing 1/2 lb of salt per gallon is poured into the tank at a rate of 2 gal/min, and the mixture is allowed to leave at the same rate. After 10 min the process is stopped, and fresh water is poured into the tank at a rate of 2 gal/min, with the mixture again leaving at the same rate. Find the amount salt in the tank at the end of an additional 10 min.
2018-02-15-9-59-55-png.png

How can the two highlighted part obtain?

Homework Equations


y(t)=(1/μ(t)) ∫ {from to to t} [yoμo+∫(g(t)μ(t))dt]

The Attempt at a Solution


I have tried to substitute the formula, S1(t)=e^(-0.02t) * ∫ {from 0 to t} (S1(t)/50)e^(0.02t) dt
which seems wrong...

Thanks
 

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  • #2
Did you try inserting the expression for the integrating factor and checking the equation explicitly? You should find that this gives you back the ODEs for ##S_1## and ##S_2##, respectively.
 
  • #3
yecko said:
S1(t)=e^(-0.02t) * ∫ {from 0 to t} (S1(t)/50)e^(0.02t) dt
integrating factor: e^(0.02t)
but S1(t) and t both have to integrate while I do not actually know what is S1(t), how can I integrate it?
thanks
 
  • #4
Your formula is not correct. It should not contain ##S_1(t)## in the integral. See the general form of the solution in the other thread.
 

Related to ODE problem -- Find the amount salt in the tank as water flows through it

1. How is the amount of salt in the tank affected by water flow?

The amount of salt in the tank is directly affected by the rate of water flow. As water flows through the tank, it carries and disperses the salt particles, causing the overall salt concentration in the tank to decrease.

2. What factors influence the rate of change of salt concentration in the tank?

The rate of change of salt concentration in the tank is influenced by several factors, such as the initial salt concentration, the rate of water flow, and the size and shape of the tank. Other factors, such as temperature and pressure, may also play a role.

3. How can I calculate the rate of change of salt concentration in the tank?

To calculate the rate of change of salt concentration in the tank, you will need to use an Ordinary Differential Equation (ODE) model. This model takes into account the initial salt concentration, the rate of water flow, and other relevant factors to determine the rate of change over time.

4. How can I use an ODE model to find the amount of salt in the tank at a specific time?

An ODE model can be used to find the amount of salt in the tank at a specific time by plugging in the known values for the variables and solving the equation. This will give you the salt concentration at that particular time, which can then be used to determine the total amount of salt in the tank.

5. Are there any limitations to using an ODE model for this problem?

While an ODE model is a useful tool for calculating the amount of salt in a tank as water flows through it, it may not account for all factors that can affect the system. For example, it may not take into account the effects of turbulence or mixing within the tank. Additionally, the accuracy of the model may depend on the accuracy of the initial data and assumptions made.

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