In the lab with small vessels and good mixers, it is a good assumption that mixing is close to perfect. In large vessels such as production fermenters, mixing is far from ideal. When a solution is added suddently to a big vessel, even with powerful, large mixers, it may take a minute or more before the added material is blended well and no differences in concentration can be measured from one location in the vessel to another.
We analyze systems by balancing mass. In other words, the rate of change in the mass of any consitiuent is the rate of addition minus the rate of leaving ± its rate of reaction. The reaction may deplete the ingredient as in the case of a nutrient that is consumed by the organisms, but the mass of organisms increases from this feeding to make their reaction term plus. Something with no reaction such as sugar when no organisms are present has no reaction term.
There need not be a term for rate of addition is a mass balance equation. For example, the microorganisms in an industrial fermentation are usually held aseptically. That means no entry of foreign organisms. For the time period during which the system is being analyzed, there probably is no addition term for organisms because they were inoculated at the start. For a continuous fermentation system, nutrients are added and all terms in the nutrient mass balances must be considered. For the organisms in a typical aseptic continuous process, the input term is zero. This results in the equation:
but the F x
term is zero. Note that the units are the same for each term (this must always be so for an
equation). The first term units are volume times concentration in mass per time-volume. The
second and third term units are flow in volume per time and concentration in mass per volume.
The last term is rate coefficient (units are 1 over time) times concentration (in mass per
volume) times volume. Thus each term's units cancel to mass / time.
The mass balance for growth-limiting nutrient is:
Where M is a coefficient for maintenance, the respiration of cells that are merely surviving.
Now there are two reaction terms, one for uptake of nutrient to grow cells and another for uptake merely to keep the cells functioning. As before the units for each term will reduce to mass / time.