One-dimensional flow models

In the flow models these effects are confined to the boundary layers, maintaining the vaHdity of the qua si-one-dimensional flow model. The flow is... 

Matrix flow relative to the reinforcing fibers is caused by thermal expansion of the fiber-matrix mass within the confines of the die and by the geometrical constriction of the die taper. Once the matrix flow distribution is known, the matrix pressure distribution may be determined using a flow rate-pressure drop relationship. One-dimensional flow models of thermoset pultrusion have been reasonably well verified qualitatively [15-17]. A onedimensional flow model of thermoplastic pultrusion [14,18] has similarly been compared with experimental data and the correlation found to be encouraging [19]. 

One-dimensional flow models are adopted in the early stages of model development for predicting the solids holdup and pressure drop in the riser. These models consider the steady flow of a uniform suspension. Four differential equations, including the gas continuity equation, solids phase continuity equation, gas-solid mixture momentum equation, and solids phase momentum equation, are used to describe the flow dynamics. The formulation of the solids phase momentum equation varies with the models employed [e.g., Arastoopour and Gidaspow, 1979 Gidaspow, 1994], The one-dimensional model does not simulate the prevailing characteristics of radial nonhomogeneity in the riser. Thus, two- or three-dimensional models are required. 

Tortuosity is a measiue of the extent to which the path traversed by fluid elements deviates from a straight-line in the direction of overall flow and may be defined as the ratio of the average length of the flow paths to the distance travelled in the direction of flow. Though the tortuosity depends on voidage and approaches imity as the voidage approaches unity, it is also affected by particle size, shape and orientation in relation to the direction of flow. For instance, for plate like particles, the tortuosity is greater when they are oriented normal to the flow than when they are packed parallel to flow. However, the tortuosity factor is not an intrinsic characteristics of a porous medium and must be related to whatever one-dimensional flow model is used to characterise the flow. 

The presence of shock waves during rapid expansion implies two seemingly insurmountable difficulties for any one-dimensional flow model. First, the assumption that the flow through a given cross section can be represented by one particular state has been shown to be highly flawed. Second, even if the flow pattern could be subdivided into zones of purely subsonic and purely supersonic flow, one-dimensional models offer no help in determining the location and shape of the various shock waves. 

Lee, C.Y. and Tallmadge, JA. (1976) Meniscus shapes in withdrawal of flat sheets. 3. Quasi-one-dimensional flow model using a stretch boundary condition. Ind. Eng. Chem. Fundam., 15 (4),... 

The core fluid density is determined from the hydrodynamic equations of continuity and motion, in conjunction with the equation of state for the fluid. In most studies to date, a one-dimensional flow model is assumed, gas effects are neglected, the core tank is considered to be rigid, and the core inlet fluid velocity is considered to be constant. The equation for the change in fluid density is then... 

The general features of two-dimensional flow with evaporating liquid-vapor meniscus in a capillary slot were studied by Khrustalev and Faghri (1996). Following this work we present the main results mentioned in their research. The model of flow in a narrow slot is presented in Fig. 10.16. Within a capillary slot two characteristic regions can be selected, where two-dimensional or quasi-one-dimensional flow occurs. Two-dimensional flow is realized in the major part of the liquid domain, whereas the quasi-one-dimensional flow is observed in the micro-film region, located near the wall. 

A steady homogeneous model is often used for bubbly flow. As mentioned previously, the two phases are assumed to have the same velocity and a homogeneous mixture to possess average properties. The basic equations for a steady one-dimensional flow are as follows. 

Crowley, C. J., G. B. Wallis, and J. J. Barry, 1992, Validation of One-Dimensional Wave Model for the Stratificd-to-Slug Regime Transition, lnt. J. Multiphase Flow 18 249 271. (3)... 

Equations 12.7.48 and 12.7.39 provide the simplest one-dimensional mathematical model of tubular fixed bed reactor behavior. They neglect longitudinal dispersion of both matter and energy and, in essence, are completely equivalent to the plug flow model for homogeneous reactors that was examined in some detail in Chapters 8 to 10. Various simplifications in these equations will occur for different constraints on the energy transfer to or from the reactor. Normally, equations 12.7.48 and 12.7.39... 

Two common types of one-dimensional flow regimes examined in interfacial studies Poiseuille and Couette flow [37]. Poiseuille flow is a pressure-driven process commonly used to model flow through pipes. It involves the flow of an incompressible fluid between two infinite stationary plates, where the pressure gradient, Sp/Sx, is constant. At steady state, ignoring gravitational effects, we have... 

The dispersion model is one of the frequently used models. It describes the dispersion of the residence time of the phases according to Fig. 9.16, for example, in one-dimensional flows by superimposing the plug profile of the basic flow with a stochastic dispersion process in axial direction, which is constructed by analogy to Pick s first law of molecular diffusion ... 

Plug Flow with Dispersion - Plug flow with dispersion is a concept that is often used to describe one-dimensional flow systems. It is somewhat more flexible in computational models because the mixing within the system is not dependent on reactor size, as with complete mix tanks in series. Plug flow with dispersion will be described in the second half of this chapter because special techniques are needed for the analysis. 

To quantify this treatment of migration as influenced by kinetics, a model has been developed in which instantaneous or local equilibrium is not assumed. The model is called the Argonne Dispersion Code (ARDISC) ( ). In the model, adsorption and desorption are treated independently and the rates for adsorption and desorption are taken into account. The model treats one dimensional flow and assumes a constant velocity of solution through a uniform homogeneous media. 

This expression describes the fastest and most important mode of transport in groundwater. In fact, an important task of the hydrologist is to develop models to predict the effective velocity u (or the specific flow rate q). Like the Darcy-Weis-bach equation for rivers (Eq. 24-4), for this purpose there is an important equation for groundwater flow, Darcy s Law. In its original version, formulated by Darcy in 1856, the equation describes the one-dimensional flow through a vertical filter column. The characteristic properties of the column (i.e., of the aquifer) are described by the so-called hydraulic conductivity, Kq (units m s"1). Based on Darcy s Law, Dupuit derived an approximate equation for quasi-horizontal flow ... 

For plug flow, only the flow and the processes other than mixing, diffusion, and conduction are considered. These have been studied in Chapter 4. In a plug flow tubular reactor model we consider only the convective one-dimensional flow and the chemical reaction as shown in Figure 5.1, where n is the convective molar flow rate for the constant volumetric flow rate g of component i. These two rates are connected by the equation rq = q Ci for the concentration Cj. 

D. Levi-Hevroni, A. Levy, I. Borde, Mathematical modelling of drying of liquid/solid slurries in steady sate one-dimensional flow, Drying Technol. 13 (5-7) (1995) 1187-1201. 

This approach is based on the similarity between a TCC and a SMB unit, such that the flow rates in a TCC can be converted easily to the equivalent ones in a SMB unit. In the frame of equilibrium theory - that is, a model assuming one-dimensional flow - adsorption equilibrium between solid and liquid phase and neglecting axial dispersion, the following mass balances are obtained for each component i in every section j of a TCC unit ... 

The heat-transfer coefficient is a function of all chain, where heat flow passes, and for one-dimensional (plane model) heat exchange between liquid and hydride bed divided by heat-conducting wall can be expressed by ... 

Although analyses in which some of these assumptions are removed, for example, by accounting for finite gas-phase reaction rates [53], for non-uniform particle-size distributions [54], [55], [56] and for droplet breakup [57], [58], are more realistic in that correlations with observed rocket-motor performance sometimes can be obtained, they involve numerical integrations which may tend to obscure the essential ideas. Simple analytical results have, however, been 5eveloped for a model that accounts in an approximate way for nonuniform size distributions [59]. Comprehensive reviews of related studies may be found in [58] and [60]. The major drawback to all of these analyses is the one-dimensional flow approximation, which excludes from consideration the three-dimensional flows generally observed in real engines. 

Remark 4.8 The results of Theorem 4.3 depend crucially on the model (the Oldroyd model). It would be interesting to know what happens for one dimensional flows of general differential models with a Newtonian contribution. 

They subsequently (2) developed a one-dimensional mathematical model in the form of coupled differential and integro-differential equations, based on a gross mechanism for the chemical kinetics and on thermal feedback by wall-to-wall radiation, conduction in the tube wall, and convection between the gas stream and the wall. This model yielded results by numerical integration which were in good agreement with the experimental measurements for the 9.53-mm tube. For this tube diameter, the flows of unbumed gas for stable flames were in the turbulent regime. 

Simple one-dimensional reservoir models (two phase Darcy flow) indicate that, in general, the flow rates across permeable fault seds will be too high to sustain high pressure gradients or corresponding differences in hydrocarbon column lengths over geo-... 

CHEMFLO Unsaturated zone CHEMFLO is a one-dimensional flow and transport model designed to simulate the movement of water and chemicals into and through soils. 

There are several theoretical approaches to the canopy flows considered by meteorologists, [17, 155, 187, 522], In the simplest one-dimensional mathematical model, the canopy is of an infinite extent along the axis Ox of the flow direction. The following mathematical model with coupled ordinary differential equations is the usual approximation in this case [155] ... 

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