A Component Balance

In this example we have a tank where a dilute salt stream is mixed with a diluent stream. We want to model how the outlet salt concentration will change when the inlet flows or inlet concentration changes. To simplify things, I am going to make a few initial assumptions. These are:

• The density of all the streams are constant and equal - this assumption is only close to valid if the streams are all very dilute - high concentrations of dissolved material will significantly change density. The problem in modelling this is that it requires physical property relationships to be included in the model, and this makes things quite complicated.

• Perfect level control - this says that the level (and hence the volume) in the tank stays constant, regardless of changes in the flows. Taken along with the constant density assumption gives us a constant overall mass balance, and hence one less equation to solve. Later in this exercise we'll work out the model if this assumption is not made.

• The tank is perfectly mixed - the salt leaving the tank is at the same concentration as the salt in the tank.

By making the perfect level control and constant density assumptions we are assuming a constant mass hold up. This means the mass flowrate out (and hence volumetric flowrate, since constant and equal densities are assumed) has to equal the total flowrate in. This means that volumetric flowrate Fout has to equal the sum of F₁ and F_d. The only balance we now need to do is a component mass balance on the salt stream.

Have a go at working out the accumulation term:

Now construct the complete component balance:

Now we can have a go at producing the model when we don't make the perfect level control assumption. In this case the volume is free to change when any of the flows in the system change and the outlet flow is now completely independent of the input flowrates.

The first thing we need to do is to derive the total mass balance for the system. Have a go at doing this yourself before looking at my answer (remember that densities are constant and equal across all the streams).

Now we can construct the component mass balance. This balance will be different from the one above because the volume can no longer be assumed constant. You need to use the product rule to separate the terms in the derivative. You can then substitute the equation arising from the total mass balance to complete the model. Have a go at doing this before revealing the solution (NB: this is a bit more difficult than the previous exercises).

This is an interesting result! It arises because the concentration in the tank is decided by the ratio of the salt to diluent stream. This kind of result is NOT general across multi-equation systems!!