Variable density model

Theodorou DN. Variable-density model of polymer melt surfaces, structure and surface tension. Macromolecules 1989 22 4578 589. 

While thin polymer films may be very smooth and homogeneous, the chain conformation may be largely distorted due to the influence of the interfaces. Since the size of the polymer molecules is comparable to the film thickness those effects may play a significant role with ultra-thin polymer films. Several recent theoretical treatments are available [136-144,127,128] based on Monte Carlo [137-141,127, 128], molecular dynamics [142], variable density [143], cooperative motion [144], and bond fluctuation [136] model calculations. The distortion of the chain conformation near the interface, the segment orientation distribution, end distribution etc. are calculated as a function of film thickness and distance from the surface. In the limit of two-dimensional systems chains segregate and specific power laws are predicted [136, 137]. In 2D-blends of polymers a particular microdomain morphology may be expected [139]. Experiments on polymers in this area are presently, however, not available on a molecular level. Indications of order on an... 

For fluid particles that continuously coalesce and breakup and where the bubble size distributions have local variations, there is still no generally accepted model available and the existing models are contradictory [20]. A population density model is required to describe the changing bubble and drop size. Usually, it is sufficient to simulate a handful of sizes or use some quadrature model, for example, direct quadrature method of moments (DQMOM) to decrease the number of variables. 

If this model is further simplified by considering unidirectional flow, the number of equations is reduced to four (Wallis, 1969). Another example is Bankoff s variable-density, single-fluid model for two-phase flow (Bankoff, 1960). Since it is based on an intimate mixture, both mechanical equilibrium (i.e., same velocity) and thermal equilibrium (same temperature) between the two phases must logically be assumed (Boure, 1975). 

Bankoff, S. G., 1960, A Variable-Density, Single-Fluid Model for Two-Phase Flow with Particular Reference to Steam-Water Flow, Trans. ASME, J. Heat Transfer 52 265-272. (3)... 

The classical CRE model for a perfectly macromixed reactor is the continuous stirred tank reactor (CSTR). Thus, to fix our ideas, let us consider a stirred tank with two inlet streams and one outlet stream. The CFD model for this system would compute the flow field inside of the stirred tank given the inlet flow velocities and concentrations, the geometry of the reactor (including baffles and impellers), and the angular velocity of the stirrer. For liquid-phase flow with uniform density, the CFD model for the flow field can be developed independently from the mixing model. For simplicity, we will consider this case. Nevertheless, the SGS models are easily extendable to flows with variable density. 

Note that these expressions (Eqs. 156 and 157) appear deceptively simple (i.e., as if the problem can be reduced to modeling a variable-density, single-phase flow) because we have hidden the difficult terms in the definition of some new symbols First, the phase-average stress a is defined by... 

Sanders, J. P. H. and I. Gokalp (1998). Scalar dissipation rate modelling in variable density turbulent axisymmetric jets and diffusion flames. Physics of Fluids 10, 938-948. 

Wang, Y. Wang, C. Y. Comparing variable density with constant density models for polymer electrolyte fuel cells. ECEC Technical Report 2003-02-, Penn State University University Park, PA, 2003. See also Modeling polymer electrolyte fuel cells with large density and velocity changes. J. Electrochem. Soc., in press. 

In the case of A as limiting reactant and a variable-density system, the solution of the model is the same as for first-order irreversible homogeneous reactions of the form A - products (Levenspiel, 1972) ... 

These results led us to analyze the relationship between carrier-wave frequency and power density. We developed a mathematical model (6) which takes into account the changes in complex permittivity of brain tissue with frequency. This model predicted that a given electric-field intensity within a brain-tissue sample occurred at different exposure levels for 50-, 147-, and 450-MHz radiation. Using the calculated electric-field intensities in the sample as the independent variable, the model demonstrated that the RF-induced calcium-ion efflux results at one carrier frequency corresponded to those at the other frequencies for both positive and negative findings. In this paper, we present two additional experiments using 147-MHz radiation which further test both negative and positive predictions of this model. 

Kumar, S. Mathematical modelling of natural convection in fire—A state of the art review of the field modelling of variable density turbulent flow. Fire and Materials, 1983. 7, 1-24. 

In the matrix model (Jongschaap, 1990), the global thermodynamic system is composed of two separate physical parts, which are called the environment and the internal variables. For the polymer solutions, for example, the pressure tensor Pv may be the internal variables, and the classical variables density, velocity, and internal energy are the environment variables. 

W. P. Jones, Models for Turbulent Flows with Variable Density and Combustion, in Prediction Methods for Turbulent Flows, W. Kollmann, ed.. New York Hemisphere Publishing Corp., 1980, 379-421. 

The error of an observation in the nth event of a sequence may be modeled as a random variable e . For a continuous random variable with range —oo.oo), a commonly used probability density model is the normal error curve... 

We assume constant-density systems for the sake of simplicity, even though variable density could be considered by a straightforward extension of this model. 

Chemical reactions may also affect turbulence by releasing energy and modifying the fluid properties locally. The influence can be quite significant in variable density flows (e.g. combustion). Nevertheless, in many computational models of constant density reactive flow processes, it is implicitly assumed that chemical reactions do not affect scalar mixing rates. 

Jones WP, Launder BE (1972) The Prediction of Laminarization with a Two-Equation Model of Turbulence. Int J Heat Mass Transfer 15 301-314 Jones WP (1980) Models for Turbulent Plows with Variable Density and Combustion. In Kollman W (ed) Prediction Methods for Turbulent Plow, Hemisphere Publishing Company... 

A problem arises with the models of Section 9.1.1 when the density of the fluid changes appreciably with temperature or pressure. To understand the difficulty, refer to Section 3.1, where variable-density PFRs are treated. We consider only the case where the tube cross section is constant. A more general version of Equation 9.1 is... 

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