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17calculus > differential equations > mixing and concentration |
Topics You Need To Understand For This Page
Differential Equations Alpha List
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Mixing and Chemical Concentration in Fluids |
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On this page we discuss one of the most common types of differential equations applications of chemical concentration in fluids, often called mixing or mixture problems. The idea is that we are asked to find the concentration of something (such as salt or a chemical) diluted in water at any given time. Usually we are adding a known concentration to a tank of known volume. Sometimes the tank is being drained at the same time. There are many variations to this problem but they are usually easily set up and solved using basic differential equations techniques. |
Basic Equation |
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There is only one basic equation you need and setting it up is the tricky part. We highly recommend that you carry your units with you as you set it up. Sometimes, mistakes can be caught when the units don't work out. The equation you need is
\(\displaystyle{ \frac{dA}{dt} = ( rate~in ) - ( rate~out ) }\) |
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1. Since this is a rate problem, the variable of integration is time t.
2. \(A\) is the amount or quantity of chemical that is dissolved in the solution (usually in water), usually with units of weight like kg.
3. The rates ( rate in and rate out ) are the rates of inflow and outflow of the chemical. The units are usually weight per unit time, like kg/min.
As you work these problems, you will notice that there are two rates involved, the rate of flow of chemical and the rate of flow of the fluid. It is important that you understand the difference when you are working these problems. We will separate the flow rates like this.
chemical-in = rate of chemical coming into the tank with units weight/time, like kg/min.
fluid-in = rate of the fluid entering the tank with units volume/time, like litres/min = L/min.
[ similarly for flow rates out of the tank ]
Notice in the table above, the rate in and rate out are chemical-in and chemical-out rates. Rewriting that equation we have
\(\displaystyle{ \frac{dA}{dt} = ( \textit{ chemical-in } ) - ( \textit{ chemical-out } ) }\).
The tricky part is knowing how to set up the rates of chemical-in and chemical-out. The chemical-in rate is usually the easiest, so let's look at that first.
Chemical-In Rate |
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Usually we are given the concentration of the fluid coming in and the rate at which it is flowing in. For example, one of the
practice problems gives the rate in as 10L/min of pure water (with no chemical or salt). There is no chemical in the solution (since it is pure water), so the amount of chemical is 0kg/L. The rate of inflow of the chemical is modeled as
\( \textit{ chemical-in } = (0 kg/L) (10 L/min) = 0 kg/min \).
Notice that we wrote our units in this equation, so that we were easily able to see that the units L canceled, leaving the units kg/min, which is what we want.
Chemical-Out Rate |
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Okay, so now let's look at outflow. This is usually where the variable we call \(A(t)\), or just \(A\), comes in. The variable \(A\) is the amount (in weight, like kg) of the chemical. It is dependent on t, so it is changing. The concentration in the tank at any time t is just \(A\) divided by the volume of fluid in the tank. The volume is calculated using this equation.
volume of fluid in the tank at any time = initial amount + ( rate coming in - rate going out ) times t
Notes
1. We assume the rates coming in and going out are constant. If they are not, the above equation would need to be adjusted accordingly.
2. If the fluid is coming in and going out at the same rate, the volume of fluid in the tank is constant and equal to the initial amount.
So, for example, if we have a tank that starts out with 1000L and fluid is coming in at a rate of 10L/min and going out at a rate of 12L/min, we have
volume of fluid in the tank = 1000L + (10L/min - 12L/min)t = 1000-2t L
This equation describes the amount of fluid in the tank at any time t and this is what is divided into A to get the rate of chemical out of the tank.
Study Tip |
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As you work these problems, it may help to set up a table containing everything you know and everything you need to find. In the first practice problem, we set one up as an example. As you study, try different formats for the table until you come up with one that makes sense for you. |
Well-Mixed Tank |
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When we discussed the outflow of the tank above, we had to use the idea that the tank was well mixed and that the chemical was dispersed evenly throughout the tank. If we didn't have this assumption, we would have needed a term in the equation that was a function of time describing how the chemical was dispersed in the tank. This is an extremely complicated concept and one that you will probably not come across in differential equations.
Before working some practice problems, let's watch a quick video explaining these types of problems in a bit more detail. This is pretty theoretical but it will give you another perspective that may be helpful.
Commutant - well-mixed tank | |
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Practice Problems |
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Instructions - - Solve the following problems, giving your answers in exact form. Unless otherwise stated, assume the tanks are well-stirred.
Level A - Basic |
Practice A01 | |
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A tank contains 1000L of fluid and 15kg dissolved salt. Fresh water enters at 10L/min and the tank is draining at 10L/min. How much salt is in the tank at t minutes and after 20 minutes? | |
solution |
Practice A02 | |
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A tank with 200 gallons of brine solution contains 40 lbs of salt. A concentration of 2 lb/gal is pumped in at a rate of 4 gal/min. The concentration leaving the tank is pumped out at a rate of 4 gal/min. How much salt is in the tank after 1 hour? How much salt is in the tank after a very long time? | |
solution |
Practice A03 | |
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A tank initially holding 80gals of fluid with 40lbs of salt has two sources of fluid being pumped in. One source is pumping pure water in at 8 gal/min and the second source is pumping in a salt solution of 0.5lbs/gal at 2 gal/min. The tank is emptying at 10gal/min. Find the amount of salt in the tank at any time t. | |
solution |
Practice A04 | |
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A 100 gallon tank with 50 gals of pure water has a solution of 0.4lbs/gal being pumped in at a rate of 3 gals/min. The well-mixed solution leaves the tank at a rate of 2 gals/min. Find an expression for the salt concentration at any time t. | |
solution |
Practice A05 | |
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A 1000L tank starts out with 200L of fluid containing 10g/L of dye. Pure water is poured in at 20 L/min and the tank is being drained at a rate of 15L/min. Write the equation for the amount of dye in the tank at any time t. | |
solution |
Level B - Intermediate |
Practice B01 | |
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Initially a tank contains 1kg of salt dissolved in 100L of water. Salty water containing 1/4kg/L at a rate of 3L/min is added to the tank and the (stirred) solution is draining from the tank at 3L/min. Determine an equation for how much salt is in the tank at any time t. | |
solution |
Practice B02 | |
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A tank holds 300 gallons of brine solution with 40 lbs of salt. A concentration of 2 lbs/gal is pumped in a rate of 4 gal/min. The concentration leaving the tank is pumped out at a rate of 3gal/min. How much salt is in the tank after 12min? | |
solution |