Reduction, generalized degree

The (generally complex) quantity p plays a role analogous to the order parameter familiar from phase transitions. The fact that only one such parameter survives in the final equations illustrates the enormous reduction of degrees of freedom associated with the first bifurcation. Note also the similarity between equations (8) and (9) and the normal forms at which one arrives in the qualitative theory of differential equations in the vicinity of resonance points.3... 

The next difficulty in using the energy balance is that thermodynamic property data, such as the heat of combustion, may not be known for some identified molecular species compounds, and is not likely to been measured for incompletely defined species such as the average biomass CHi.sOo.sNo.t- To estimate the properties in these cases, we will use the generalized degree of reduction, - c. defined as follows ... 

The average biomass used to represent many yeasts and bacteria has an elemental composition, on a C-moIe basis, of CH. sOo.5No.2. Compute its generalized degree of reduction, and estimate its Gibbs energy and heat of combustion. 

It is useful to compare this answer with the more approximate one that is obtained using the generalized degree of reduction for all components ... 

There is one linear combination of the carbon, nitrogen, and oxygen balances that is especially useful. If we take the sum of four times the carbon balance plus the hydrogen balance and subtract twice the oxygen balance (note that the multipliers are the valences, and are the same numbers that appear in the definition of the generalized degree of reduction), we obtain... 

It is of interest to examine the simplified version of the second-law constraint in terms of the generalized degrees of reduction. In this case we have... 

To begin, we compute the generalized degree of reduction of the biomass produced ... 

Each of these modes is assigned to a generalized degree of freedom (q-set). The accuracy of the dynamic reduction step is determined by the number of retained normal modes nq. As the number of generalized degrees of freedom Uq needed for an accurate description of the dynamic behaviour of the component is usually several orders smaller than the number of internal dofs, the size of the component model is drastically reduced. 

Table 4 Kharasch and generalized degree of reduction definition for ammonia and biomass.

Calculation of the degree of reduction and generalized degree of reduction was presented in equations (22) and (25), respectively. The low value of the atomic coefficient of sulfur in biomass has a low influence on the degree of reduction. 

Table 9 Elemental analysis of microorganisms. Mass fractions are expressed as g per 100 g biomass. Mx is the mass of one C-mole of biomass, and y the generalized degree of reduction. The standard deviation is indicated in parenthesis [37,38]. 

---