Single-phase flow equations

If the obvious equalities in the single-phase flow equations above are given separate designations, then these quantities can be compared to their counterparts when two-phase flow equations are written in the same form. Therefore, in Eq. (18), and (19), we define the following terms ... 

For single-phase flows, pressure is shared by three momentum equations and requires special algorithms to compute the pressure field. Most of these algorithms (discussed in the previous chapter) use one continuity equation and three momentum equations to derive pressure and/or pressure correction equations. However, for multiphase flows, there is more than one continuity equation. Answers to questions such as which continuity equation should be used to derive pressure equations are not obvious. As discussed in the previous chapter, it is customary to employ iterative techniques to solve single-phase flow equations. Such iterative techniques can, in principle, be extended to simulate multiphase flows. In practice, however, the process... 

SINGLE-PHASE FLOW Equations for Fluid Motion... 

If, due to pressure drop and temperature rise, the liquid hydrogen is permitted to reach the saturation line at a point preceding the required design length, then a two-phase flow condition will exist which will cause the single-phase flow equations to be invalid. Due to added turbulence, the frictional pressure drop for two-phase flow will tend to be greater than a pressure drop for a singlephase system with equal mass flow rate. 

For isotropic homogeneous porous media (uniform permeability and porosity), the pressure for creeping incompressible single phase-flow may be shown to satisfy the LaPlace equation ... 

In the above equations, a is a coefficient with the value of 1.0 for single phase flow and 2.0 for multi-phase flow [6], and pis an adjustable coefficient and has a value of 2.1 by fitting the experimental results for the two phase flow. The flow resistant coefficient is determined by the Blasius equation. 

Single-phase flow region at the inlet the liquid is below its boiling point (sub-cooled) and heat is transferred by forced convection. The equations for forced convection can be used to estimate the heat-transfer coefficient in this region. 

The number of equations to be solved is, among other things, related to the turbulence model chosen (in comparison with the k-e model, the RSM involves five more differential equations). The number of equations further depends on the character of the simulation whether it is 3-D, 21/2-D, or just 2-D (see below, under The domain and the grid ). In the case of two-phase flow simulations, the use of two-fluid models implies doubling the number of NS equations required for single-phase flow. All this may urge the development of more efficient solution algorithms. Recent developments in computer hardware (faster processors, parallel platforms) make this possible indeed. 

The CFD models considered up to this point are, as far as the momentum equation is concerned, designed for single-phase flows. In practice, many of the chemical reactors used in industry are truly multiphase, and must be described in the context of CFD by multiple momentum equations. There are, in fact, several levels of description that might be attempted. At the most detailed level, direct numerical simulation of the transport equations for all phases with fully resolved interfaces between phases is possible for only the simplest systems. For... 

That is, the two-phase frictional pressure gradient is calculated from a reference single-phase frictional pressure gradient (dP/dx)R by multiplying by the two-phase multiplier, the value of which is determined from empirical correlations. In equation 7.73 the two-phase multiplier is written as < >% to denote that it corresponds to the reference single-phase flow denoted by R. 

By analogy to single-phase flow, the two-phase frictional pressure-drop can be expressed by the conventional Fanning equation, and thereby a friction factor is defined. These friction factors may be based on liquid properties, gas properties, or on a fictitious single fluid of mean properties obtained by some averaging procedure. Typical definitions, such as those shown in equation (32), have been given and discussed recently by Govier and Omer (G4) ... 

Computational fluid dynamics enables us to investigate the time-dependent behavior of what happens inside a reactor with spatial resolution from the micro to the reactor scale. That is to say, CFD in itself allows a multi-scale description of chemical reactors. To this end, for single-phase flow, the space resolution of the CFD model should go down to the scales of the smallest dissipative eddies (Kolmogorov scales) (Pope, 2000), which is inversely proportional to Re-3/4 and of the orders of magnitude of microns to millimeters for typical reactors. On such scales, the Navier-Stokes (NS) equations can be expected to apply directly to predict the hydrodynamics of well-defined system, resolving all the meso-scale structures. That is the merit of the so-called DNS. 

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 conservation equations of mass, momentum, and energy of a single-phase flow can be obtained by using the general conservation equation derived previously. 

In order for a model to be closured, the total number of independent equations has to match the total number of independent variables. For a single-phase flow, the typical independent equations include the continuity equation, momentum equation, energy equation, equation of state for compressible flow, equations for turbulence characteristics in turbulent flows, and relations for laminar transport coefficients (e.g., fJL = f(T)). The typical independent variables may include density, pressure, velocity, temperature, turbulence characteristics, and some laminar transport coefficients. Since the velocity of gas is a vector, the number of independent variables associated with the velocity depends on the number of components of the velocity in question. Similar consideration is also applied to the momentum equation, which is normally written in a vectorial form. 

At this point, a review is in order. Please note again Eq. (6.6). The Darcy equation, also known as the Fanning equation, is applicable to any single-phase flowing fluid, liquid or gas. There are seven factors that make up the dimensional analysis of this equation ... 

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