Tag: unsteady state

In my previous post I explained (briefly) what mass balancing was. Here I will go over the rules and equations used for mass balancing.

Mass Balancing: General equation

The most important equation that covers all aspects of mass balancing is this:

[Mass in + mass generated] – [Mass out + Mass consumed] = Mass accumulated

Pretty self-explanatory, right? Mass in is the amount of what you put into the system, mass generated is the mass of anything generated inside the system, mass out is the mass of what leaves the system, mass consumed is the mass of what is consumed in the reactor and mass accumulated is the mass of anything that is still inside the system.
This equation can apply to the whole unit operation (the mass of everything all together) or to individual components (water, nutrients, product).

However, depending on the type of process or material you have, you can simplify the equation.

Change in Process variables

A fermentation can be split into 2 states:

• Steady state: In this state, process variables do not change with time. This means that there is also no change in mass for each component over time. This state generally applies to continuous fermentation, where input, output and the system are the same over time.
• Unsteady state: Process variables change over time. This is the case for batch or semi-batch operations, where material is not added or removed from the reactor at a constant rate. Due to this, the mass of different components may differ at different stages of the fermentation.

Depending on the state of the process, the mass balance equation will change!

Mass Balance: Simplified equations

Steady state

Since mass does not change over time, there is no accumulation of mass within the reactor/ system. Therefore, the equation can be simplified to these terms:

Mass in + Mass generated = Mass out + Mass consumed

If there are no reactions occurring in a steady state process, meaning the molecules in that component do not interact with each other (i.e. cell mass), there is no mass generated or consumed. Then, the equation can further be simplified to:

Mass in = Mass out

Both these equations can be applied to either the process as a whole (if the above is applicable) or to individual components.

Unsteady state

In an unsteady state process, we should use the general equation. However, we tend to assume (at least at the bachelor degree level) there is no accumulation by assuming total conversion of input to output. Why? This is simpler, and in a batch or semi-batch process the unconverted nutrients tends to be a very small value. We also only do a mass balance of the whole process, not random time points, since this is more useful and much easier.

Therefore, if we assume no accumulation:

Mass in + Mass generated = Mass out + Mass consumed

No reaction

For some components of the process, there is no reaction occurring between them and other components, resulting in no accumulation, generation or consumption. This applies to elements (hydrogen, oxygen, nitrogen) since they can’t be broken down. It can also apply to other components but always double check that they aren’t reactive! In this case, the amount entering should be the same as the amount leaving, in which case:

Mass in = Mass out

Overall process

Mass cannot be created or destroyed, it can only be transferred. Because of this, for the whole process, regardless of whether its steady, unsteady, batch or continuous, the mass of everything entering the reactor will always be the same as the mass of everything leaving the reactor. This is because reactors are closed systems, and do not allow free exchange of mass in or out of the vessel. Therefore, the mass balance for the whole process will be:

MASS IN = MASS OUT

Those are all the rules! My advice would be to use the general equation for each component, and if there is no accumulation/ consumption / generation, just cross them out of the equation. Usually though, the only equations you need are:

MASS IN = MASS OUT

for components with no reaction, and

MASS IN + MASS GENERATED = MASS OUT + MASS CONSUMED


for components with reactions

If you want to know whether you can use the mass in = mass out equation, you can use this guide:

Does mass in = mass out?
Material	Without reaction	With reaction
Total mass	Yes	Yes
Total number of moles	Yes	No
Mass of the total amount of molecules	Yes	No
Number of moles of a molecule	Yes	No
Mass of an element	Yes	Yes
Number of moles of an element	Yes	Yes

This is complicated and hard to explain with words, so I will do some working out of example questions in later posts to help you understand it better. However, if you have any questions about these rules, please leave a comment and I will get back to you ASAP!