74417586_2648405288580195_7902770812829040640_n.jpg - 19 MIXTURE PROBLEMS 67 Procedure for Flow Problems 1 It is helpful to draw a rough sketch of a

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Unformatted text preview: 19 MIXTURE PROBLEMS 67 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. FIGURE 1.9.1 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. outflow. 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 pumped in. 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 5 my/sec indicating the inflow and the outflow (see Figure 1.9.2). Water flows in at the conc. = rate of 5 m'/s, with concentration 0.15 g/m', and the mixture flows out at 0.15 g/m 5.me/sec 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 FIGURE 1.9.2 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 1.9.1. inflow and outflow, so that the rate of change of the amount of pollutar...
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  • Fall '10
  • capretta

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