Unstructured models

Unstructured model uncertainty relates to unmodelled effects such as plant disturbances and are related to the nominal plant CmCv) as either additive uncertainty (s)... 

When microbial cells are incubated into a batch culture containing fresh culture media, their increase in concentration can be monitored. It is common to use the cell dry weight as a measurement of cell concentration. The simplest relationships describing exponential cell growth are unstructured models. Unstructured models view the cell as an entity in solution, which interacts with the environment. One of the simplest models is that of Malthus 19... 

Models for batch culture can be constructed by assuming mechanisms for each phase of the cycle. These mechanisms must be reasonably comph-cated to account for a lag phase and for a prolonged stationary phase. Unstructured models treat the cells as a chemical entity that reacts with its environment. Structured models include some representation of the internal cell chemistry. Metabolic models focus on the energy-producing mechanisms within the cells. 

This section gives models for the rates of birth, growth, and death of cell populations. We seek models for (1) the rate at which biomass is created, (2) the rates at which substrates are consumed, (3) the rates at which products are generated, (4) the maintenance requirements for a static population, and (5) the death rate of cells. The emphasis is on unstructured models. 

At each temperature the simple Monod kinetic model can be used that can be combined with material balances to arrive at the following unstructured model... 

Stirred tank reactor (ST/ ). The differential mass balance referred to the azo-dye converted by bacteria (assuming unstructured model for the biophase, i.e., that it is characterized only by cell mass or concentration X) yields... 

Continuous Stirred Tank Reactor (CSTR). The conversion degree of the azo-dye, the reaction volume (V) and the volumetric flow rate (Q) of the dye-bearing stream are related to each other through the material balance referred to the dye and extended to the reactor volume. Assuming an unstructured model for the biophase, the material balance yields... 

The mass balance on dye in a STR operated batchwise assuming an unstructured model yields... 

Aerobic The growth kinetics was described by an interacting, balanced and unstructured model characterized by phenol inhibition and oxygen limitation according to a double limiting kinetics [60, 62],... 

When a more detailed analysis of microbial systems is undertaken, the limitations of unstructured models become increasingly apparent. The most common area of failure is that where the growth is not exponential as, for example, during the so-called lag phase of a batch culture. Mathematically, the analysis is similar to that of the interaction of predator and prey, involving a material balance for each component being considered. 

Even though Williams s model provides many features that unstructured models are unable to predict, it requires only two parameters, which is the same number of parameters required for Monod kinetics. 

Certainly, Monod s formula has been used extensively in phenomenological (unstructured) models, although the literature presents other equations for one limiting substrate systems (Equations 17 and 18). In Moser s formulation it was necessary to introduce a third parameter ( n in Equation 17) to represent experimental data. 

Unstructured models, as detailed in Sections 8.3.1 and 8.3.2, are formulated by a series of kinetic and differential non-linear equations that represent the dynamics of all the state variables during the process. Thus, to simulate a model that consists of parameters and state variables, it is necessary to attribute values to the parameters. 

In Section 4, competition between two populations is analyzed. Again, the equations can be reduced to a system that can be directly compared to the systems derived in Chapters I and 2. Section 5 explores the evolution in time of the population average length, surface area, and volume in Section 6 we formulate the conservation principle, which played such a crucial role in earlier chapters. The steady-state size distribution of a population is determined in Section 7. Our findings are summarized in a discussion section, where a comparison is made between the conclusions derived from the size-structured model and the unstructured models considered in Chapters 1 and 2. 

One can distinguish between structured and unstructured models, the latter neglecting intracellular phenomena. On the contrary, structured models account for intracellular processes and states in different compartments of the cell or include explicit kinetics for various intracellular steps of virus replication. 

Despite the high social relevance of infectious diseases and widespread use of animal cell lines in vaccine production, the application of even unstructured models for quantitative analysis and parameter estimation has not been common practice in bioprocess optimization. So far, research concerning influenza vaccine production in MDCK cell cultures has focused on the characterization of metabolism, growth of different cell lines and virus yields in various production systems [1,2]. 

Figure 1 shows the evolution of virus yields over time for two of the three observed initial conditions. There is no detectable difference between the present structured model and the unstructured approach of Mohler et al. [4]. That is because of the unstructured model being included in the structured model as the zeroth order moment. 

Figure 1. Outer dynamics in comparison with model of Mohler et al. [4] virus yield vs. time post infection for different MOI (circles experimental results, solid line unstructured model, dashed line structured model, tshift = 4.5 h)...

A new deterministic population balance model with distributed cell populations has been presented. The model is based on the unstructured approach of Mohler et al. [4]. Concerning outer dynamics the present model is equivalent to the unstructured model which proved to be sufficient to predict virus yields for different initial conditions [4]. The characteristics of the inner dynamics can be simulated except of the decrease of fluorescence intensity at later time points. The biological reasons for this effect are unclear. Presumably there are more states that have to be considered during virus replication like intercellular communication, extent of apoptosis or specific stage in cell cycle. Future computational and experimental research will aim in this directions and concentrate on structured descriptions of the virus replication in mammahan cell culture. 

Where S, G, X, E and Enz are respectively the starch, glucose, cells, ethanol and enzyme concentrations inside the reactor, Si is the starch concentration on the feed, F is the feed flow rate, V is the volume of hquid in the fermentor and (pi, (p2, (ps represent the reaction rates for starch degradation, cells growth and ethanol production, respectively. The unstructured model presented in (Ochoa et al., 2007) is used here as the real plant. The ki (for i=l to 4) kinetic parameters of the model for control were identified by an optimization procedure given in Mazouni et al. (2004), using as error index the mean square error between the state variables of the unstructured model and the model for control. 

Figure 3. Soft Sensors Vs Real Plant (Unstructured Model)...

The control scheme shown in Figure 4 was applied to the SSFSE process using the fed batch version of the unstructured model proposed in Ochoa et al. (2007) as the object for control. Simulations of starch and glucose concentrations are corrupted by additive noise e. These white noise signals, simulate measurement noises at 5% of the standard deviation for the mean values of both S and G concentrations. As stated before, the control law (17) will be applied only when the marker is negative therefore, the control algorithm block is expressed as follows ... 

Unstructured models view the cell as a single component interacting with the fermentation medium, and each bioreaction is considered to be a global reaction, with a corresponding empirical rate expression. 

Additional complexity can be included through cell population balances that account for the distribution of cell generation present in the fermenter through use of stochastic models. In this section we limit the discussion to simple black box and unstructured models. For more details on bioreaction systems, see, e.g., Nielsen, Villadsen, and Liden, Bioreaction Engineering Principles, 2d ed., Kluwer, Academic/Plenum Press, 2003 Bailey and Ollis, Biochemical Engineering Fundamentals, 2d ed., McGraw-Hill, 1986 Blanch and Clark, Biochemical Engineering, Marcel Dekker, 1997 and Sec. 19. 

Reusing parts of VeDa and of the VeDa-based implementation models of the afore mentioned research prototypes ROME and ModKit (cf. Sect. 5.3), the partial model Mathematical Models comprises both generic concepts for mathematical modeling and specific types of mathematical models. Within the partial model Unstructured Models, models are described from a mathematical point of view. Concepts of Unstructured Models can be used to specify the equations of System Models, which model Systems in a structured manner. Models for the description of ChemicalProcessSystems (as defined in CPS Models) are examples for system models. Such models employ concepts from Material Model to describe the behavior of the materials processed by the chemical process system. Finally, the partial model Cost Models introduces methods to estimate the cost for construction, procurement, and operation of chemical process systems. 

The implementation model of ModKit- - [151] is based on the partial models Unstructured Models and System Models. 

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