Unformatted text preview: 19 MIXTURE PROBLEMS
Procedure for Flow Problems
1. It is helpful to draw a rough sketch of a tank illustrating the inflow and
outflow with pipes (see Figure 1.9.1).
2. Label quantities and note the given data.
3. Express inflow rates and outflow rates in terms of the given variables and
substitute them into (1).
Tank with inflow and
4. Solve the resulting differential equation.
5. Answer any questions such as "how long?"
We will discuss problems of two different kinds. In one, the volume of
water in the tank will be fixed, and the resulting differential equation will be
easier to solve. In the other, the volume of water will be changing in time,
and the resulting differential equation will be harder to solve. In some of the
exercises, the process of setting up the differential equations will be emphas
sized, and you will not be asked to solve the differential equations.
1.9.1 Mixture Problems with a Fixed Volume
We begin with a fixed volume example.
EXAMPLE 1.9.1 Fixed Volume
Consider a 100-m' tank full of water. The water contains a pollutant at a
concentration of 0.6 g/m . Cleaner water, with a pollutant concentration of
m', is pumped into the well-mixed tank at a rate of 5 m /s. Water
flows out of the tank through an overflow valve at the same rate as it is
a. Determine the amount and concentration of the pollutant in the tank as a
function of time. Graph the result.
b. At what time will the concentration be 0.3 g/m'?
SOLUTION In order to illustrate the general principles, and since this is our
first mixing problem, we shall include a few more steps than are necessary to
solve the particular problem.
In mixture problems, it is best to first draw a rough diagram of a tank
indicating the inflow and the outflow (see Figure 1.9.2). Water flows in at the
rate of 5 m'/s, with concentration 0.15 g/m', and the mixture flows out at
the same rate of 5 m /s. Thus, the volume of water in the tank stays the
same, equaling 100 m'. Usually, it is easier to formulate a differential
equation for the amount of the pollutant. We let Q() be the amount in
grams of pollutant in the tank. @ depends on the time , which we measure in
Picture for Example
seconds. The amount of pollutant in the tank changes in time as a result
inflow and outflow, so that the rate of change of the amount of pollutar...
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- Fall '10