Network-based constitutive model

Knowledge based approaches such as fuzzy logic, neural networks or multiagents model currently constitute an important axis of research and application in bioprocesses. They have shown their usefulness particularly when one does not have an analytical model but that a certain expertise is available. Harmand and Steyer [37] have addressed that when this expertise comprises a sufficiently important know-how, approaches such as fuzzy logic will be preferred. If, on the other hand, one has only a limited experience but lays out of a rather important data base, the statistical approaches such as neural networks can be used. 

Heinrich, G., Kaliske, M., 1997. Theoretical and numerical formulation of a molecular based constitutive tube-model of rubber elasticity. CompuL Theor. Polym. Sci. 7 (3-4), 227-241. Heinrich, G., Straube, E., Helmis, G., 1988. Rubber Elasticity of Polymer Networks Theories Polymer Physics. Springer, Berlin/Heidelberg, pp. 33-87. 

Sperling has studied theoretical conditions for the formation of domains in sequential IPNs using cross-linking degree for each network, as well as thermodynamics of mixing and interfacial tension for sequential IPNs, where separation occurs by the nucleation mechanism. The derivation of the basic equation for IPN domain diameters is based on a physical model of sequential IPNs, according to which polymer II, which is formed in a swollen network I, constitutes a spherical core and is in a contracted (deformed) state, while polymer I surrounds the core and is in an expanded (deformed) state. 

With roughly 1000 atoms, the size of the silicon clusters that constitute the micro PS network is between the bulk crystal and a molecule. Hence models of the luminescence process based on size reduction of the crystal, as well as models based on molecular sttuctures, have been proposed, which are reviewed in detail in [Ca7, Ju3]. Generally the various models of the luminescence of PS can be classified into three major categories ... 

The relaxing Gaussian network of Green and Tobolsky (4) is the earliest version of this model. Lodge (12) and Yamamoto (J5) independently derived constitutive equations for similar systems, based on a stress-free state for each newly created strand and a distribution of junction lifetimes which is independent of flow history. For Gaussian strands in an incompressible system ... 

Process-scale models represent the behavior of reaction, separation and mass, heat, and momentum transfer at the process flowsheet level, or for a network of process flowsheets. Whether based on first-principles or empirical relations, the model equations for these systems typically consist of conservation laws (based on mass, heat, and momentum), physical and chemical equilibrium among species and phases, and additional constitutive equations that describe the rates of chemical transformation or transport of mass and energy. These process models are often represented by a collection of individual unit models (the so-called unit operations) that usually correspond to major pieces of process equipment, which, in turn, are captured by device-level models. These unit models are assembled within a process flowsheet that describes the interaction of equipment either for steady state or dynamic behavior. As a result, models can be described by algebraic or differential equations. As illustrated in Figure 3 for a PEFC-base power plant, steady-state process flowsheets are usually described by lumped parameter models described by algebraic equations. Similarly, dynamic process flowsheets are described by lumped parameter models comprising differential-algebraic equations. Models that deal with spatially distributed models are frequently considered at the device... 

Some researchers have used approximate microscopic descriptions to develop more rigorous macroscopic constitutive laws. A microstructural model of AC [5] linked the directionality of mechanical stiffness of cartilage to the orientation of its microstructure. The biphasic composite model of [6] uses an isotropic fiber network described by a simple linear-elastic equation. A homogenization method based on a unit cell containing a single fiber and a surrounding matrix was used to predict the variations in AC properties with fiber orientation and fiber-matrix adhesion. A recent model of heart valve mechanics [8] accounts for fiber orientation and predicts a wide range of behavior but does not account for fiber-fiber interactions. 

Two different constitutive equations, namely the Wagner model and the Phan Thien Tanner model, both based on network theories, have been investigated as far as their response to simple shear flow and uniaxial elongational flow is concerned. This work was primarily devoted to the determination of representative sets of parameters, that enable a correct description of the experimental data for three polyethylenes, to be used in numerical calculation in complex flows. Additionally, advantages and problems related to the use of these equations have been reviewed. 

The shape of the capillary portion of the liquid-vapor interfacial area for sand (Fig. 1-1 lb) resembles simulation results of Reeves and Celia (1996) of interfacial areas in pore networks due to capillarity only. The discussion illustrates potential limitations in using cylindrical pore network models (Reeves Celia, 1996) especially for studies of volatile liquids and surfactants, and other multiphase transport processes where interfacial areas play a crucial role (Kim et al., 1997 Karkare Si. Fort, 1996). Furthermore, the overwhelming role of adsorbed liquid films casts doubts on several proposed constitutive relationships between capillary pressure (saturation) and interfacia] area (Skopp, 1985 Hassanizadeh Gray, 1993) most of which were based on assumed cylindrical capillary geometry in the absence of adsorption. 

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