provide the necessary information about the features of the chosen bioprocessing system; (c) it synthesizes the characteristics of the specified living cells’ evolution and hence, it is the best technique to predict the process efficiency.

The models show the complex biosystems attributes; so they must be as possible as extensive and non-speculative. Moreover the models are an acceptable compromise between the presentation of processes in detail, with considerable number of parameters, and the use of few parameters, easy to apply and estimate.

Most important properties of a biological mathematical model were defined in the Edwards and Wilke’ postulates [2]: (a) it is capable to represent all the culture phases; (b) it is flexible enough to approximate different data types without the insertion of significant distortions; (c) it must be continuously derivable; (d) it must be easy to operate, once the parameters evaluated; (e) each model parameter is to have a physic significance and must be easy to evaluate.

The attempts to realize high global models were not successful: firstly, due to the impossibility to measure on-line the great number of bioprocess parameters, and secondly, due to the high degree of complexity. Finally several types of models can represent the evolution of the aerobic bioprocess. The most important categories will be presented further on.

1. The unstructured global models are in use nowadays as the main tool for both the bioprocess modeling, but also for being applied in overall computer control [2]. Their limit is they are a simplified representation of the bioprocess behavior: conforming to this concept the bioprocess evolution depends directly and only on the macroscopic variables representing the working conditions in the bioreactor. Therefore the unstructured models are essentially kinetic equations that describe the variation of substrate or product concentrations and of a unique biological state variable-the cell concentration, and can also express the influences of some important process variables (pH, pO2, temperature, and others), and only sometimes they are balance equations.

1 dX

Generally speaking [9], one considers that the specific growth rate (ц =———— ) is the key

X dt

variable for cell growth, substrate consumption and product formation. The specific growth rate is time dependent and dependent on different physical, chemical and/or biological parameters (substrate concentration-S, cell concentration-X, product concentration-P, pH, temperature-T, dissolved oxygen concentration-C, and different inhibitors-I).

Conforming to the literature assumptions [10], the specific growth rate dependence upon different process parameters can be considered as follows:

ц = f (S, X, P, pH, C, I,….,t) (1)

a. p=p(S) Kinetic models with growth limitation through substrate concentration (without inhibition) Main model equations [2, 11] are presented in Table 1.

Model equation	Constants	Authors	Comments
U(S) = U4 (2) KS + S	pmax=max specific growth rate [1/h] KS = saturation constant [g/L]	Monod equation (1942, 1949)	Empirically derived from the Michaelis & Menten equation
u(S) = UmaxS (3) Ks + S"		Moser equation (1988)	Analogy with a Hill kinetic (n>0)
S u(S) = Umax KS + KD + s ^4)	KD=diffusion constant	Powell equation (1958)	Influence of cell permeability, substrate diffusion and cell dimensions through KD parameter

Table 1. Models u=u(S)

There are also some models, which utilize the substrate concentration in more complex structures. Nyholm (1976) introduces a dual function for substrate utilization: consumption (including assimilation and dissimilation in the liquid phase) and growth (substrate utilization for growth):

Se is the substrate for growth and Sa the substrate used for consumption. The growth rate is linked to the intracellular concentration of limiting substrate (Snt/X) and to preserved substrates (i. e. inorganic ions or vitamins, not decomposed through cell metabolism) with application in wastewater bio treatment:

b. U=u(X, S) The influence of cell and substrate concentrations upon the specific growth

rate2′ 11

Model equation	Constants	Authors	Comments
UY) = Umax(1 — KY (7)	kX=kinetic constant	Verhulst (1845)	It is known as growth logistic model
So — X U(X, S) = Umax——— YY (8) Ks + So — Y	So=substrate initial concentration Y=substrate/cell yield.	Meyrath (1973)	It is based on Monod kinetics.

N — N0 exp(Pmaxt) No¥zxeMu’Lj) (9) pax + mxN0(eMprnJ) — 1)	N=population density m=limiting size of the population (the carrying capacity)	Verhulst — Pearl kinetics	Logistic growth: combination between the population trend to growth according to a geometric progression and the environment tendency to limit the excessively high densities of the population
KxX +S (M)	KX=kinetic constant	Contois (Contois — Fujimoto) equation (1959):	If S = constant, the only dependence remains p = f(X).

Table 2. Modelsp =p(X, S)

c. Growth kinetics with substrate inhibition

In most cases, the kinetic model equations are derived (like the Monod model) from the inhibition theory of enzymatic reactions. Consequently they are not generally valid and can be applied in connection with experimental acceptability [2, 11].

Model equation	Constants definition	Authors name	Comments
p — p 1 = S 1 (11) P Pmax1 + Ks + S Ks + S 1 + S_ s К k	Ki = inhibition constant	Andrews model (1968)	Substrate inhibition in a chemostat
s S(1+¥) p — p Ks (12) r* rmax c2 S + k^s¥t kS	Ksl= inhibition constant	Webb model (1963)
P-Pmax К s (13) 1+—+У(—І s Yk ¥	Ki, S= inhibition constant	Yano model (1966)
S p — p —S—e K’,s (14) P Pmax ks + S		Aiba model (1965)

Table 3. Growth kinetics with substrate inhibition

d. p = f(S, P) Growth kinetic with product inhibition [2, 11]

Hinshelwood (1946) detected product inhibition influences upon the specific growth rate: linear decrease, exponential decrease, growth sudden stop, and linear/exponential decrease
in comparison with a threshold value of P. The first type (Hinshelwood — Dagley model):

P) =ftmax (1 — kP) (15)

KS + S

where: k = inhibition constant (considering the product concentration influence). Model equation Constants definition Authors name M(P) =^max — Kt(P — K2) (16) K1, K2 = constants (>0) Holzberg model (1967) P( P) =^max(1 P ) 1 max (17) Pmax = maximum product concentration. Ghose and Tyagi model (1979) M( P) = ^maxe ) (18) K1 = constant Aiba (1982): ^P) =Mmax kS+ Se"P (19) Aiba and Shoda model (1989) Table 4. Models^ = f(S, P)

e. The influence of dissolved oxygen (as a second substrate) upon the specific growth rate

In some cases it is needed to consider the dissolved oxygen as a second substrate. The most used equation is the kinetic model with double growth limitation, p(S, C) [2, 11]

i. Olsson model:

where: KC = oxygen saturation constant.

ii. Williams’ model, which also quantifies the P influence (Kp=P saturation constant; Kt, K2, K3, K4=modeling constants):

f. |i(St, S2) Kinetic models based on different substrates

Besides the case when the dissolved oxygen is considered as a second substrate, there are many cases when two or more carbon sources are taken into consideration. There are two typical situations: (1) the cells grow through the sequential (consecutive) substrate consumption (diauxic growth), where a simple Monod model can be applied; (2) the cells grow through the simultaneous consumption of substrates (e. g. wastewater treatment); in this case, the mathematical modeling is more complex.