Noncalculability of Stoichiometry

Suppose that in Example 1a the chosen two conversion rates to be measured are biomass production (rx) and

NH+ consumption (rN). Measurement of these two rates would not lead to a calculation of the other rates because rX and rN occur in the nitrogen-conservation equation in such a way that rX uniquely determines rN, and vice versa. It is then said that rX and rN are redundant. The N balance gives a constraint for these two measured conversion rates that can be used to calculate the statistically best estimate of rX and rN, which also exactly satisfies the N balance.

Clearly the choice of the two measured rates must be such that calculability of all other conversion rates is assured. In example 1a, suitable combinations would be the oxygen consumption rate (rO) and biomass production rate (/X), rO and the carbondioxide production rate (rC), or rx and the heat production rate (/q).

Ill-Conditioned Calculability of Stoichiometry (Error Propagation)

It is well known that measured conversion rates have a certain measurement error. The subsequently calculated conversion rates, from combining the conservation relations and the two measurements, have an error due to error propagation. It is obviously of great practical importance to choose two measured conversion rates where this error propagation is minimal. A simple example to illustrate this problem is the aerobic growth of biomass on the donor glucose. If oxygen consumption ( — rO) and carbon dioxide production (rC) are the measured rates, then the following relations (using conservation relations and the standard biomass composition) to calculate rX and — rD (in C-mol glucose/m3 h) from the measured rC and (— rO) can be derived:

rx = 20rc — 20( —rQ) ( —rD) = 20( —rQ) — 21 rc

Due to the large multiplication factors of 20 and 21 in these equations, the propagation of the measurement errors in rC and rO into rX and — rD is enormous.

If the donor conversion rate (— rD) and the carbon dioxide production rate (rC) were chosen as the measured rates rX and (— rO) would be calculated as dundant measurements, which can be used for two purposes error diagnosis and data reconciliation.

Error Diagnosis

• To check the validity of the defined growth systems with respect to the absence of by-products or possible second substrates

• To check the measured conversion rates for systematic errors

Data Reconciliation

• To decrease the measurement error in the calculated and measured conversion rates, provided that the statistically based checks (error diagnosis) on the validity of the growth system and the systematic errors in the measured conversion rates are passed 