## Example

Use of redundant measurements to establish the presence of errors in the definition of the growth system or in the measurements

Consider the microbial growth system of Example 1a, where the following four conversion rates have been measured. The biomass has the standard elemental composition.

Biomass production O2 consumption

HCO3 production Oxalic acid (C2O;i —

consumption)

We know that a minimum of two rate measurements are needed to calculate the full stoichiometry. Therefore there are two redundant measurements. We can now establish the conservation equations (with rW, rH, and rN as the water, proton, and NH^ conversion rates, respectively) based on conversion rates as

C conservation H conservation O conservation N conservation Charge conservation

The error propagation now is much lower and, therefore, from the measured ( — rD) and (rC), rX and ( —rO) can be calculated, as can be the other conversion rates involved. Clearly the aspect of error propagation is of major importance, and this propagation can be significantly decreased by a proper choice of the conversion rates to be measured.

### Redundancy of Measurements

As stated earlier, in general two well-chosen measured conversion rates are usually sufficient to reliably calculate the complete stoichiometry. However, it is advantageous (Example 2) to measure more conversion rates than the minimum requirement of two. This leads to so-called re-

By eliminating the three nonmeasured rates (rW, rH, rN) from these five conservation equations, one obtains 5 — 3 = 2 equations, which relate the measured conversion rates only. The result is as follows:

The first relation can be recognized as the carbon balance, and the second is so-called electron balance or the balance of degree of reduction (1) (see also a later section). With respect to the C balance, one finds from the measurements:

C-in = 2 x 5.8(oxalate) = 11.6 C-mol/h C-out = 1(biomass) + 10.5(CO2) = 11.5 C-mol/h

Clearly the C balance seems satisfying (0.86% gap).

For the balance of degree of reduction one obtains electrons in = 2 X 5.8 = 11.6 mol electrons/h electrons out = —1.2(—4) + 1(4.2) = 9 mol electrons/h

Clearly there is a large gap of 2.6 mol electrons/h.

Because the C balance fits, it is reasonable to assume that the measured values of rD, rX, and rC are reliable. The balance of degree of reduction can therefore be wrong for two reasons:

### 1. A very inaccurate measurement of rO.

2. If the measurement of rO is found to be correct, then the only other possibility is the presence of an additional electron acceptor (e.g., NO—). This would be an error in the defined growth system.

A Mathematically Complete Analysis of Calculability, Analysis of Redundancy, Error Diagnoses, and Data Reconciliation

In the preceding section simple examples were provided to highlight the problems in accurately establishing the full growth stoichiometry from measurements. Because all these calculations are based on linear conservation relations, it is highly appropriate to use matrix algebra. Basic to these calculations is the "elemental" matrix, which specifies the element, charge, and enthalpy information for each compound in the growth system. Recently, an extensive and coherent mathematical description has been provided for calculability, redundancy analysis, error diagnosis, statistical aspects, and data reconciliation using involved matrix algebra (18-21). The developed mathematical theory has been put in a user-friendly computer program called Macrobal (22).

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