# 1.2 The mass balance: examples

Course subject(s) 1. Balance Equation: our working horse

## EXAMPLE 1.2A: DILUTING A TOXIC WATER SUPPLY (ELEMENTARY)

A continuous stirred tank (CSTR) has a volume of 5 liters. It contains 0.5 liters of dirty water with a concentration c0 = 100 gram/liter of some toxic substance represented by variable X. At some moment, 2.5 liters of clean water is added.

1. Calculate the concentration of X after adding the fresh water.
2. In order to be able to clean this water using micro organisms, the concentration of X needs to be lower than 0.1 gram/liter. Is it possible to add a sufficient amount of water to the tank such that the concentration of X is below this limit?

## Example 1.2A: Diluting toxic water supply (elementary)

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## EXAMPLE 1.2B: DILUTING WITHIN A STIRRED TANK REACTOR (MEDIUM)

A continuous stirred tank is filled with water that contains kitchen salt at a concentration of 30 kg/m3. The water volume is 100 liter. From t=0 onwards, fresh water flows in at a flow rate of 5 l/s. The flow rate out of the tank is the same.

1. What is the salt concentration if t goes to infinity?
2. Set up a salt mass balance for the stirred tank.
3. Solve this balance and find what the concentration in the salt is at t=20 seconds.

## Example 1.2B: Diluting within a stirred tank reactor (medium)

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## EXAMPLE 1.2C: ANTIBIOTICS IN A BODY (ADVANCED)

The human kidney removes all kinds of molecules from our body. This includes medicine, that you might receive in your blood, during a serious illness. This could be an antibiotic, e.g. Gentamicin. In this exercise, we are going to investigate how long it takes to remove 50% of the used Gentamicin. We assume that the intake is instantenous and that your blood has a concentration of c(0)= c0 of Gentamicin.
Our kidneys do not process our entire blood stream during one cycle. They roughly process 20% of the total blood stream. They don’t remove all Gentamicin that flows through them; actually only a small fraction. We can model this by stating that a blood flow of 0.5 ml/sec is going through our kidneys and is completely cleaned from Gentamicin.
We can treat our blood system as a CSTR and our heart as the pump (of negligible volume) that pumps the blood around. For the removal of the Gentamicin, 0.5 ml/sec of blood flows through our kidneys and gets cleaned. For the modelling of this system, treat the blood system as a 5 liter CSTR.

1. Make a sketch of the blood-kidney system.
2. Set up a mass balance for the blood system, treating it as a CSTR. The mass to be considered is that of Gentamicin. At t=0, the concentration of Gentamicin is c0.
3. Solve this mass balance and compute how long it takes before the concentration of Gentamicin is half of c0.

## Example 1.2C: Antibiotics in a body (advanced)

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## BONUS: EXAMPLE 1.2D: UNSTEADY STATE MASS BALANCE (MEDIUM)

Let’s consider a vessel where a certain mass flow rate of an aqueous salt solution (cs,in= 1.0 g/L) enters a well-defined volume and the same amount leaves the volume. Initially, the volume was filled with a salt solution with a concentration of 5 g/L water. The vessel is well mixed by a stirrer.

What is the concentration after 20 seconds in the vessel if the volume flow rate is equal to 10 L/s and the volume of the vessel is 100 L?

## Bonus: Example 1.2D: Unsteady state mass balance (medium)

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The maths in this course contains only some elements of calculus, but sometimes mistakes are made.

In order to help you, Peter Hamersma talks about some common mistakes made by his students. Don’t worry if you don’t get the last one. Everything will get clearer during the course.