Continuum flow models

Holt et al. [16] measured water and gas flow through the pores of double-walled carbon nanotubes. These tubes had inner diameters less than 2 nm with nearly defect-free graphitic walls. Five hydrocarbon and eight non-hydrocarbon gases were tested to determine flow rates and to demonstrate molecular weight selectivity compared with helium. Water flow was pressure driven at 0.82 atm and measured by following the level of the meniscus in a feed tube. The results for both gas and liquid show dramatic enhancements over flux rates predicted with continuum flow models. Gas flow rates were between 16 and 120 times than expected according to the Knudsen diffusion model in which fluid molecule-wall collisions dominate the flow. 

Molecular and continuum flow models. (From Gad-el-Hak, M., Flow physics, in The MEMS Handbook, Gal-el-Hak, M., Ed., CRC Press, Boca Raton, FL, 2002, p. 4-3. Originally from Gad-el-Hak, M. (1999) /. Fluids Eng. 121, pp. 5-33, ASME, New York. With permission.)... 

Regarding the transport of the liquid water, the flow rate observed by the experiment can not be explained by the continuum flow models. The experimental permeation is more than three orders of magnitude faster than the value predicted by the Poiseuille equation. In such a small pore whose diameter is about seven water molecules, the velocity profile, inherent for the continuum theory, seems difficult to apply. The ejqierimentally observed water flux compares well with the value predicted by the molecular dynamic (MD) model [2]. The simulation predicts the flow... 

The above results show close agreement between the experimental and theoretical friction factor (solid line) in the limiting case of the continuum flow regime. The Knudsen number was varied to determine the influence of rarefaction on the friction factor with ks/H and Ma kept low. The data shows that for Kn < 0.01, the measured friction factor is accurately predicted by the incompressible value. As Kn increased above 0.01, the friction factor was seen to decrease (up to a 50% X as Kn approached 0.15). The experimental friction factor showed agreement within 5% with the first-order slip velocity model. 

Pressure drop and heat transfer in a single-phase incompressible flow. According to conventional theory, continuum-based models for channels should apply as long as the Knudsen number is lower than 0.01. For air at atmospheric pressure, Kn is typically lower than 0.01 for channels with hydraulic diameters greater than 7 pm. From descriptions of much research, it is clear that there is a great amount of variation in the results that have been obtained. It was not clear whether the differences between measured and predicted values were due to determined phenomenon or due to errors and uncertainties in the reported data. The reasons why some experimental investigations of micro-channel flow and heat transfer have discrepancies between standard models and measurements will be discussed in the next chapters. 

It should be pointed out that the flow rate in the case of the Couette flow is independent of the inverse Knudsen number, and is the same as the prediction of the continuum model, although the velocity profiles predicted by the different flow models are different as shown in Fig. 4. The flow velocity in the case of the plane Couette flow is given as follows (i) Continuum model ... 

Notwithstanding the natural heterogeneity of the subsurface, we can usefully consider homogeneous (bulk, effective) descriptions for at least some problems, especially for water flow (but less so for contaminant migration see Sect. 10.1). Therefore, two basic approaches to modeling generally are used to describe and quantify flow and transport continuum-based models and pore-network models. We discuss each of these here. 

Abstract In this contribution, the coupled flow of liquids and gases in capillary thermoelastic porous materials is investigated by using a continuum mechanical model based on the Theory of Porous Media. The movement of the phases is influenced by the capillarity forces, the relative permeability, the temperature and the given boundary conditions. In the examined porous body, the capillary effect is caused by the intermolecular forces of cohesion and adhesion of the constituents involved. The treatment of the capillary problem, based on thermomechanical investigations, yields the result that the capillarity force is a volume interaction force. Moreover, the friction interaction forces caused by the motion of the constituents are included in the mechanical model. The relative permeability depends on the saturation of the porous body which is considered in the mechanical model. In order to describe the thermo-elastic behaviour, the balance equation of energy for the mixture must be taken into account. The aim of this investigation is to provide with a numerical simulation of the behavior of liquid and gas phases in a thermo-elastic porous body. 

There have been many hybrid multiscale simulations published recently in other diverse areas. It appears that the first onion-type hybrid multiscale simulation that dynamically coupled a spatially distributed 2D KMC for a surface reaction with a deterministic, continuum ODE CSTR model for the fluid phase was presented in Vlachos et al. (1990). Extension to 2D KMC coupled with ID PDE flow model was described in Vlachos (1997) and for complex reaction networks studied using 2D KMC coupled with a CSTR ODEs model in Raimondeau and Vlachos (2002a, b, 2003). Other examples from catalytic applications include Tammaro et al. (1995), Kissel-Osterrieder et al. (1998), Qin et al. (1998), and Monine et al. (2004). For reviews, see Raimondeau and Vlachos (2002a) on surface-fluid interactions and chemical reactions, and Li et al. (2004) for chemical reactors. 

In the Multigrain model, fractured catalyst microparticles are produced during the polymerization and uniformly dispersed in the polymer each of these particles behaves as a micro Solid core and diffusion within them, as well as in the interstices between them, can take place. In the Polymeric flow model the catalyst microparticles are dispersed in a polymer continuum and move outward in proportion to the volumetric expansion due to polymerization only one value of diffusivity is considered. Both these models predict significant MWD broadening due to mass transfer limitations (Q , 9 for polypropylene in the Polymeric flow model) on the basis of mathematical calculations carried out assuming reasonable values of the kinetic and physical parameters. 

The continuum scale is an intermediate range in terms of flow modelling, in that some methods are based on detailed or discrete modelling to predict average properties of dispersion, while others are based on continuum modelling approaches using aver-age/statistical descriptions of the aggregated properties of the canopy and the flow (see Table 2.7). 

MODELING FLOW IN POROUS MATERIALS Continuum-Scale Modeling... 

The two-fluid granular flow model is formulated applying the classical Eulerian continuum concept for the continuous phase, while the governing equations of the particle phase are developed in accordance with the principles of kinetic theory. In this theory it is postulated that the particulate system can be represented considering a collection of identical, smooth, rigid spheres, adapting a Boltzmann type of equation. This microscopic balance describes the rate of change of the distribution function with respect to position and time. 

Bear (1969, 1972) developed a dispersive flow model based upon the idea of building a continuum at the mesoscopic scale by statistically averaging microscopic quantities over a representative elementary volume, defined with respect to porosity. Phis geometric model is an assemblage of randomly interconnected tubes... 

Geologic materials like oil shale are commonly treated as elastic/plastic solids. Fracture under intense loading is then modeled as an extension of plasticity or is treated with a separate fracture model. In applications like rock blasting, the latter approach is preferable, since fracture of rock is qualitatively different from plastic flow. Even so, continuum damage models have been used to model blasting for engineering applications. Some calculations with such a model will be presented below. 

Migdal, D., and Agosta, V. D. "A Source Flow Model for Continuum Gas-Particle Flow." Journal of Applied Mechanics 35 (1967) 860-65. 

Figure 2 shows the basic physical idea of the microstructure of the continuum rheologicS model we proposed earlier (2). The layers can be idealized as separated by porous slabs, which are connected by elastic springs. Liquid crystals may flow parallel to the planes in the usual Newtonian manner, as if the slabs were not there. In the direction normal to the layers, liquid crystals encounter resistance through the porous medium, proportional to the normal pressure gradient, which is known as permeation. The permeation is characterized by a body force which in turn causes elastic compression and splay of the layers. Applied strain from the compression causes dislocations to move into the sample from the side in order to relax the net force on the layers. When the compression stops and the applied stress is relaxed the permeation characteristic has no influence on stress strain field. 

A broad classification of the various quasi-continuum models is presented in Table 12.1. The simplest is clearly the one-dimensional pseudohomogeneous plug-flow model (Al-a) in which the radial gradients of heat and mass within a tube are neglected. Then complicating factors can be added, one at a time, to allow for increasing reality,... 

Moholkar and Pandit (2001b) have also extended the nonlinear continuum mixture model to orifice-type reactors. Comparison of the bubble-dynamics profiles indicated that in the case of a venturi tube, a stable oscillatory radial bubble motion is obtained due to a linear pressure recovery (with low turbulence) gradient, whereas due to an additional oscillating pressure gradient due to turbulent velocity fluctuation, the radial bubble motion in the case of an orifice flow results in a combination of both stable and oscillatory type. Thus, the intensity of cavitation... 

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