Constitutive equation Differential models

There are many ways of writing equations that represent transport of mass, heat, and fluids trough a system, and the constitutive equations that model the behavior of the material under consideration. Within this book, tensor notation, Einstein notation, and the expanded differential form are considered. In the literature, many authors use their own variation of writing these equations. The notation commonly used in the polymer processing literature is used throughout this textbook. To familiarize the reader with the various notations, some common operations are presented in the following section. 

In general, the utilization of integral models requires more elaborate algorithms than the differential viscoelastic equations. Furthermore, models based on the differential constitutive equations can be more readily applied under general concUtions. 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

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... 

The three constant Oldroyd model is a nonlinear constitutive equation of the differential corrotational type, such as the Zaremba-Fromm-Dewitt (ZFD) fluid (Eq. 3.3-11). [For details, see R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Second Edition, Vol. 1, Wiley, New York, 1987, Table 7.3-2.]... 

The mathematical model of catalyst/polymer particle evolution consists of the set of differential-algebraic equations (61)-(64). The constitutive equations describing the force interactions, transport of monomer, phase equilibria at the interface between polymer and pore phase as well as the rules for connectivity of micro-elements have to be specified (Grof and Kosek, 2005 Grof et al., 2005a). 

In the various formulations of the mathematical theory of linear viscoelasticity, one should differentiate clearly the measurable and non-measurable fimctions, especially when it comes to modelling apart from the material functions quoted above, one may also define non measurable viscoelastic functions which Eu-e pure mathematical objects, such as the distribution of relaxation times, the distribution of retardation times, and tiie memory function. These mathematical tools may prove to be useful in some situations for example, a discrete distribution of relaxation times is easy to handle numerically when working with constitutive equations of the difierential type, but one has to keep in mind that the relaxation times derived numerically by optimization methods have no direct physical meaning. Furthermore, the use of the distribution of relaxation times is useless and costs precision when one wishes simply to go back and forth from the time domain to the frequency domain. This warning is important, given the large use (and sometimes overuse) of these distribution functions. 

Many improvements or modifications to the UCM model can be found in the literature. These csm lead to various classes of constitutive equations keeping the differential nature of the equation [2, 3, 35]. As pointed out by Larson [43], a systematic classification of these can be performed by rewritting the UCM model as ... 

These constitutive equations differ in their mathematical form the Wagner equation is an integral equation whereas the Phan Thien Tanner model is a differential one. 

Of course this list is not exhaustive. (See other models in [1,2].) Also models with internal variables (order parameters) as those of [3] can be put in a similar (though more complicated) framework. In particular, there are additional constitutive equations of differential type for the order parameters [4]. 

In the story of numerical flow simulation, the ability to predict observed and significant viscoelastic flow phenomena of polymer melts and solutions in an abrupt contraction has been unsuccessful for many years, in relation to the incomplete rheological characterization of materials, especially in elongation. The numerical treatments have often been confined to flow of elastic fluids with constant viscosity, described by differential constitutive equations as the Upper Convected Maxwell and Oldroyd-B models. Fortunately, the recent possibility to use real elastic fluids with constant viscosity, the so-called Boger fluids [10], has narrowed the gap between experimental observation and numerical prediction [11]. 

We present here numerical results obtained in a planar short die geometry with both constitutive equations for the two differential models ((30B and mPTT). 

Computations were performed for the long dies using the three constitutive equations. For short dies, only differential models (GOB and mPTT) were tested. 

In axisymmetric flow situations, the global pressure drop in a capillary rheometer is well described by the three constitutive equations. If one focuses on the entrance pressure drop, the numerical entrance pressure drop related to Bagley correction is foimd to be less important than the corresponding experimental data for the differential models for LDPE and LLDPE melts. For the Wagner integral constitutive equation, the computed entrance pressure drops are found to be lower for both fluids, but the computed values are closer to the experimental data for LLDPE than those related to the LDPE melt. This descrepancy, previously reported in the literature, needs further investigation. 

If we divide the airshed into L cells and consider N species, LN ordinary differential equations of the form (15) constitute the airshed model. As might be expected, this model bears a direct relation to the partial differential equations of conservation (7). If we allow the cell size to become small, it can be shown that (15) is the same as the first-order spatial finite difference representation of (7) in which turbulent diffusive transport is neglected—i.e,. 

One must note that the balance equations are not dependent on either the type of material or the type of action the material undergoes. In fact, the balance equations are consequences of the laws of conservation of both linear and angular momenta and, eventually, of the first law of thermodynamics. In contrast, the constitutive equations are intrinsic to the material. As will be shown later, the incorporation of memory effects into constitutive equations either through the superposition principle of Boltzmann, in differential form, or by means of viscoelastic models based on the Kelvin-Voigt or Maxwell models, causes solution of viscoelastic problems to be more complex than the solution of problems in the purely elastic case. Nevertheless, in many situations it is possible to convert the viscoelastic problem into an elastic one through the employment of Laplace transforms. This type of strategy is accomplished by means of the correspondence principle. 

Figure 2 shows the comparison of the fractal-layer (solid line a) and two-timescale (solid line b) models with the simulations in terms of effective diffusivity, eq. (13). Both the models furnish a satisfactory level of agreement with simulation data. We may therefore conclude that approximate models based on a Riemann-Liouville constitutive equation are able to furnish an accurate description of adsorption kinetics on fractal interfaces. These models can also be extended to nonlinear problems (e.g. in the presence of nonlinear isotherms, such as Langmuir, Freundlich, etc.). In order to extend the analysis to nonlinear cases, efficient numerical sJgorithms should be developed to solve partied differential schemes in the presence of Riemann-Liouville convolutional terms. 

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