# Simple D.E mixing tank problem

• leehufford
In summary, the conversation discusses a mixing tank problem with an initial value of 300 gallons of water and 50 lbs of salt. The differential equation for the amount of salt in the tank at time t > 0 is determined to be dA/dt + 7A/(600-t) = 6 with an initial condition of A(0) = 50. The question is raised about how to determine the correct denominator for the volume of the tank, which is found to be V=300-t/2. The concentration of salt and the rate at which salt is being lost are also considered in the conversation.
leehufford
Hello,
Having trouble with one quirk to this mixing tank problem:

All large mixing tank initially holds 300 gallons of water, with 50 lbs of salt already in it (initial value problem). A brine solution of 2 lb/gal is pumped in at 3 gallons per minute. The water is pumped out at the faster rate of 3.5 gal/min. Determine a differential equation for the amount of salt A(t) in the tank at time t > 0.

I came up with

dA/dt + 7A/600 = 6. A(0) = 50.

Now I understand intuitively that the volume of the tank is decreasing so the 600 in the denominator is changing with time. The answer is

dA/dt + 7A/(600-t) = 6. A(0) = 50

My question is how do they know to subtract exactly t. The tank is losing 0.5 gal/min. Thanks in advance,

Lee.

leehufford said:
Hello,
Having trouble with one quirk to this mixing tank problem:

All large mixing tank initially holds 300 gallons of water, with 50 lbs of salt already in it (initial value problem). A brine solution of 2 lb/gal is pumped in at 3 gallons per minute. The water is pumped out at the faster rate of 3.5 gal/min. Determine a differential equation for the amount of salt A(t) in the tank at time t > 0.

I came up with

dA/dt + 7A/600 = 6. A(0) = 50.

Now I understand intuitively that the volume of the tank is decreasing so the 600 in the denominator is changing with time. The answer is

dA/dt + 7A/(600-t) = 6. A(0) = 50

My question is how do they know to subtract exactly t. The tank is losing 0.5 gal/min. Thanks in advance,

Lee.

The volume in V in the tank is V=300-t/2, right? As you said, you are losing 1/2 gal/min. The concentration of salt is A/V. How much salt are you losing per minute if you are pumping 3.5 gal/min out?

1 person

## 1. What is a "Simple D.E mixing tank problem"?

A "Simple D.E mixing tank problem" is a mathematical problem that involves determining the concentration of a substance in a mixing tank as it is being mixed with another substance. This problem is commonly used in chemical engineering to model the behavior of mixing tanks in various industrial processes.

## 2. How is the concentration of the substance in the mixing tank determined?

The concentration of the substance in the mixing tank can be determined using a differential equation, which takes into account the rate at which the substance is being added to the tank, the rate at which it is being mixed, and the rate at which it is being removed from the tank.

## 3. What factors affect the concentration of the substance in the mixing tank?

The concentration of the substance in the mixing tank is affected by several factors, including the flow rate of the substance being added to the tank, the mixing rate, the volume of the tank, and the rate at which the substance is being removed from the tank.

## 4. Can this problem be solved using a computer program?

Yes, this problem can be solved using a computer program that can solve differential equations. Many software programs, such as MATLAB and Mathematica, have built-in solvers for differential equations and can be used to solve the "Simple D.E mixing tank problem" quickly and accurately.

## 5. How is the "Simple D.E mixing tank problem" used in real-world applications?

The "Simple D.E mixing tank problem" is often used in chemical engineering and other industrial processes to model the behavior of mixing tanks. It can help engineers optimize the design and operation of mixing tanks to achieve desired concentrations and mixing rates, leading to more efficient and cost-effective processes.

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