Our BASIC exercise allows choice of the models for a single vessel, for a cascade, and for recycle. One of the models compares one CSTR with a cascade of two CSTR's where each has half the volume of the single CSTR. By implication, if two vessels are better then one, three would be even better, etc. Note that the dilution in the cascade of total volume equal to the single CSTR is twice as great. The equations are:
Mass balance for cells
- single or first vessel: dx/dt = µ x - Dx (6)
- second vessel: dx2/dt = µ x2 - D x2 + D x (7)
Mass balance for growth-limiting nutrient
- single or first vessel: dS/dt = D So - D S - µ x/Y - M x (8)
- second vessel: dS2/dt = D S - D S2 - µ x2/Y - Em x2
The analysis of a single CSTR with recycle applies to the usual method of operating activated sludge processes for treatment of wastes, and the cells tend to be near starvation.
The equations are:
Mass balance for cells: dx/dt = µ x - Dx + Dr xr (10)
Mass balance for growth-limiting nutrient
dS/dt = D So - D S - µ x/Y - Mx + DrS (11)
Dr = dilution rate contribution from recycle
xr = cell concentration in recycle