7GConcentration


 * Concentration and Tank Problems**

This topic is related to dilution of chemicals in tanks of water.

In these questions, water is flowing into a tank where it is mixed and then water is flowing out. If the total volume remains constant, the questions are easier.

Concentration

Recall that math . \qquad \text{Concentration } = \dfrac{\text{mass of chemical}}{\text{volume of water}} math

Definition ... ... **__Brine__** is salt-water -- water with salt disolved in it.

Units ... ... If the mass of the chemical is **kg** and the volume is **m 3 **, the units for concentration will be **kg/m 3 .**

Tank Problems

Suppose that brine is flowing into a tank through one pipe. The water is stirred within the tank and the resulting brine is flowing out through a second pipe.

We use **Q** for the Quantity of salt in the tank at time, **t**.

**Then** math . \qquad \dfrac{dQ}{dt} = \text{ Rate In } - \text{ Rate Out } math

**or** math . \qquad \dfrac{dQ}{dt} = \dfrac{dQ(in)}{dt} - \dfrac{dQ(out)}{dt} math

**Rate In = (concentration of inflow) × (rate of inflow)**

math . \qquad \dfrac{dQ(in)}{dt} = C(in) \times \dfrac{dV(in)}{dt} math

**Rate Out = (concentration within tank) × (rate of outflow)**

math . \qquad \dfrac{dQ(out)}{dt} = \dfrac{Q(t)}{V(t)} \times \dfrac{dV(out)}{dt} math

Note: We always assume that the water within the tank is perfectly mixed at all times. In this way we don't have to consider rate of mixing, or layers within the tank.

Tank problems involve setting up the differential equation dQ/dt.

If the volume is changing with respect to time, the differential equation will have __**two**__ variables (**t** and **Q**).
 * As a result we can't solve it using Year 12 calculus.
 * You can (and will) be asked to set up the D.E. but not solve it using calculus.
 * You may be asked to use your calculator to solve it

We will examine methods of solving DEs like this using direction fields.0

Your calculator can solve most DEs like this.

If the volume is constant, the DE will have one variable **Q**. You are expected to be able to solve these types of DEs using calculus.

Example 1

A tank initially contains 50 litres of water in which is disolved 10kg of salt. Brine containing salt with a concentration of 2kg/litre is flowing into the tank at a rate of 5 litres/minute. The mixture is stirred continuously and flows out of the tank at a rate of 3 litres/minute. Write the differential equation for the rate of change in the quantity of salt (**Q**) at time **t**.

Solution

Example 2

A holding tank is used to control the release of pollutants into a sewerage system. The tank initially contains 400 litres of water with 800kg of pollutants disolved in it. Polluted water containing 4 kg of pollutants per litre flows into the tank at a rate of 3 litres/minute. The mixture is kept uniform by stirring and it flows out of the tank at the same rate. a) Set up the differential equation describing the rate of change of the quantity of pollutants (Q kg) in the tank. b) Solve using calculus to find an expression for Q(t) c) Find the amount of pollutants after 20 minutes.

Solution

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