Modeling of Single-Phase Flows

Arbogast, T. and Douglas, J. and Hornung, U., Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Appl. Math. 21, 823-836, 1990... 

A more rigorous viscous turbulent model of single-phase flow, based on a Prandtl mixing length theory was published by Bloor and Ingham. Like Rietema, these authors obtained theoretical velocity profiles, but they used variable radial velocity profiles calculated from a simple mathematical theory. The turbulent viscosity was then related to the rate of strain in the main flow and the distribution of eddy viscosity with radial distance at various levels in the cyclone was derived. 

The two models commonly used for the analysis of processes in which axial mixing is of importance are (1) the series of perfectly mixed stages and (2) the axial-dispersion model. The latter, which will be used in the following, is based on the assumption that a diffusion process in the flow direction is superimposed upon the net flow. This model has been widely used for the analysis of single-phase flow systems, and its use for a continuous phase in a two-phase system appears justified. For a dispersed phase (for example, a bubble phase) in a two-phase system, as discussed by Miyauchi and Vermeulen, the model is applicable if all of the dispersed phase at a given level in a column is at the same concentration. Such will be the case if the bubbles coalesce and break up rapidly. However, the model is probably a useful approximation even if this condition is not fulfilled. It is assumed in the following that the model is applicable for a continuous as well as for a dispersed phase in gas-liquid-particle operations. 

Inasmuch as the nature of pipeline elements sets these networks apart from electrical networks (more commonly referred to as electrical circuits) we shall review briefly the modeling of these elements. We shall, however, limit ourselves to the correlations developed for single-phase fluid flow the modeling of two-phase flow is a subject of sufficient diversity and complexity to merit a separate review. 

The "correlative" multi-scale CFD, here, refers to CFD with meso-scale models derived from DNS, which is the way that we normally follow when modeling turbulent single-phase flows. That is, to start from the Navier-Stokes equations and perform DNS to provide the closure relations of eddy viscosity for LES, and thereon, to obtain the larger scale stress for RANS simulations (Pope, 2000). There are a lot of reports about this correlative multi-scale CFD for single-phase turbulent flows. Normally, clear scale separation should first be distinguished for the correlative approach, since the finer scale simulation need clear specification of its boundary. In this regard, the correlative multi-scale CFD may be viewed as a "multilevel" approach, in the sense that each span of modeled scales is at comparatively independent level and the finer level output is interlinked with the coarser level input in succession. 

One of the simplified heat transfer models of two-phase flows is the pseudocontinuum one-phase flow model, in which it is assumed that (1) local thermal equilibrium between the two phases exists (2) particles are evenly distributed (3) flow is uniform and (4) heat conduction is dominant in the cross-stream direction. Therefore, the heat balance leads to a single-phase energy equation which is based on effective gas-solid properties and averaged temperatures and velocities. For an axisymmetric flow heated by a cylindrical heating surface at rw, the heat balance equation can be written as... 

The model given above is called the k- -kp model, which can be used for dilute, non-swirling, nonbuoyant gas-solid flows. For strongly anisotropic gas-solid flows, the unified second-order moment closure model, which is an extension of the second-order moment closure model for single-phase flows [Zhou, 1993], may be used. 

The particle model This model attributes pressure drop to friction losses due to drag of a particle. The preeence of liquid reduces the void fraction of the bed and also increases the particle dimensions. Ergun (94) applied this model for single-phase flow (e.g., fixed and fluidized beds). Stichlmair et al. (95) successfully extended this model to correlate pressure drop and flood for both random and structured packings. Their correlation is complex and requires some additional validation, but is the most fundamental correlation available. 

The aim of this section is to understand the features of single-phase flow in the cathode channel of a PEFC or DMFC. The model below takes into account mass and momentum transfer through the channel/GDL interface. The model gives exact solutions and helps in clarifying how the electrochemical reactions and electro-osmotic effect affect the flow in the fuel cell channels (Kulikovsky, 2001). 

Only the single-phase flow case is considered in this chapter. In general, modeling and analyses of two-phase flow NCLs are conducted in the time domain by applications of system-analysis models and computer codes, and these are based on highly detailed and complex models of two-phase flows. Analyses based on these detailed models are basically analytically intractable. 

The modeling is based on adaptation of the equations in the previous section to the coupled loops case. The model equations developed herein will be written for the case of single-phase flow in the primary and secondary loops. Both steady-state and off-normal transient conditions in the Gen IV nuclear reactor case involve two-phase fluid states. Safety-grade analyses of design and off-normal states will generally be handled by systems-analysis models and codes that easily accommodate generalized geometry, fluid states, and flow directions. 

Methods for determining the drop in pressure start with a physical model of the two-phase system, and the analysis is developed as an extension of that used for single-phase flow. In the separated flow model the phases are first considered to flow separately and their combined effect is then examined. 

The pore geometry described in the above section plays a dominant role in the fluid transport through the media. For example, Katz and Thompson [64] reported a strong correlation between permeability and the size of the pore throat determined from Hg intrusion experiments. This is often understood in terms of a capillary model for porous media in which the main contribution to the single phase flow is the smallest restriction in the pore network, i.e., the pore throat. On the other hand, understanding multiphase flow in porous media requires a more complete picture of the pore network, including pore body and pore throat. For example, in a capillary model, complete displacement of both phases can be achieved. However, in real porous media, one finds that displacement of one or both phases can be hindered, giving rise to the concept of residue saturation. In the production of crude oil, this often dictates the fraction of oil that will not flow. 

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). 

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