Two-fluid equations

Let us summarize onr progress to this point in analyzing the stability of the int ace between the snpoposed llnids of Figure 5.1. We solved the equations of continuity and motion in Section 2.1 to obtain the perturbations in the velocity and pressure distributions in the two fluids (Equations 5.17 through 5.20 and Equation 5.26). With Equation 5.42 and the equality of Pb imphed by... 

Consider stratified flow of two immiscible fluids a and b in a horizontal (or slightly inclined) conduit. The flow configuration and coordinates are described in Figure 1. Detailed derivations of the transient one-dimensional averaged two-fluid equations can be found in many studies (e.g., Yadigaroglou and Lahey [54], Hancox et al. [55], Banerjee and Chan [56], Banerjee [57,58], Andron [59], Kocamustafaogullari [60]). 

The transient formulation of the two-fluid equations requires closure laws for the local and instantaneous shear stresses. The conventional way of modelling the wall and interfacial shear stresses is by assuming quasi-steady relations, whereby T, and X. are modelled in terms of the local phases insitu holdup and velocities ... 

The dynamic component of the closure law proposed includes a dynamic coefficient, C. This coefficient has been found to depend on the liquid layer Froude and Reynolds numbers. With the proposed dynamic model for the interfacial shear stress the transient two-fluid equations are capable of predicting the conditions for the evolution of waves in a variety of two-fluid systems (without any further tuning). 

New phenomena have been predicted on the basis of the two-fluid equations of motion—for example, the existence of a novel wave mode known as second sound in which the normal and superfluid components oscillate in antiphase such that the density remains constant. The characteristic wave velocity (typically 20 m sec but depending on temperature) is much slower than that of ordinary or first sound (typically 240 m sec ), a pressure-density wave in which the two components oscillate in phase. Second sound is a temperature-entropy wave at constant pressure and density it may be created by means of an oscillating voltage applied to a heater and may be detected by means of a thermometer. 

This equation is useful for gases above the critical point. Only reduced pressure, P, and reduced temperature, T, are needed. In the form represented by equation 53, iteration quickly gives accurate values for the compressibiUty factor, Z. However, this two-parameter equation only gives accurate values for simple and nonpolar fluids. Unless the Redhch-Kwong equation (eq. 53) is expHcifly solved for pressure in nonreduced variables, it does not give accurate hquid volumes. 

For proper use of the equations, the chamber shape must conform to the spray pattern. With cocurrent gas-spray flow, the angle of spread of single-fluid pressure nozzles and two-fluid pneumatic nozzles is such that wall impingement wiU occur at a distance approximately four chamber diameters below the nozzle therefore, chambers employing these atomizers should have vertical height-to-diameter ratios of at least 4 and, more usually, 5. The discharge cone below the vertical portion should have a slope of at least 60°, to minimize settling accumulations, and is used entirely to accelerate gas and solids for entty into the exit duct. 

Equation 2-5 gives a value for U based on the outside surface area of the tube, and therefore the area used in Equation 2-3 must also be the tube outside surface area. Note that Equation 2-5 is based on two fluids exchanging heat energy through a solid divider. If additional heat exchange steps are involved, such as for finned tubes or insulation, then additional terms must be added to the right side of Equation 2-5. Tables 2-1 and 2-2 have basic tube and coil properties for use in Equation 2-5 and Table 2-3 lists the conductivity of different metals. 

In the basic heat transfer equation it is necessary to use the log mean temperature difference. In Equation 2-4 it was assumed that the two fluids are flowing counter-current to each other. Depending upon the configuration of the exchanger, this may not be true. That is, the way in which the fluid flows through the exchanger affects LMTD. The correction factor is a function of the number of tube passes and the number of shell passes. 

The fundamental idea of this procedure is as follows For a system of two fluid phases containing N components, we are concerned with N — 1 independent mole fractions in each phase, as well as with two other intensive variables, temperature T and total pressure P. Let us suppose that the two phases (vapor and liquid) are at equilibrium, and that we are given the total pressure P and the mole fractions of the liquid phase x, x2,. .., xN. We wish to find the equilibrium temperature T and the mole fractions of the vapor phase yu y2,. .., yN-i- The total number of unknowns is N + 2 there are N — 1 unknown mole fractions, one unknown temperature, and two unknown densities corresponding to the two limits of integration in Eq. (6), one for the liquid phase and the other for the vapor phase. To solve for these N +2 unknowns, we require N + 2 equations of equilibrium. For each component i we have an equation of the form... 

Via Eq. (136) the kinematic condition Eq. (131) is fulfilled automatically. Furthermore, a conservative discretization of the transport equation such as achieved with the FVM method guarantees local mass conservation for the two phases separately. With a description based on the volume fraction fimction, the two fluids can be regarded as a single fluid with spatially varying density and viscosity, according to... 

For other discussions of two-phase models and numerical solutions, the reader is referred to the following references thermofluid dynamic theory of two-phase flow (Ishii, 1975) formulation of the one-dimensional, six-equation, two-phase flow models (Le Coq et al., 1978) lumped-parameter modeling of one-dimensional, two-phase flow (Wulff, 1978) two-fluid models for two-phase flow and their numerical solutions (Agee et al., 1978) and numerical methods for solving two-phase flow equations (Latrobe, 1978 Agee, 1978 Patanakar, 1980). 

We now have five remaining unknowns—Qm, pm, p, Lm and (AP)f— and only two remaining equations, so we still have three arbitrary choices. Of course, we will choose a pipe length for the model that is much less than the 700 miles in the field, but it only has to be much longer than its diameter to avoid end effects. Thus we can choose any convenient length that will fit into the lab (say 50 ft), which still leaves two arbitrary unknowns to specify. Since there are two fluid properties to specify (p and p), this means that we can choose (arbitrarily) any (Newtonian) fluid for the lab test. Water is the most convenient, available, and inexpensive fluid, and if we use it (p = 1 cP, p = 1 g/cm3) we will have used up all our arbitrary choices. The remaining two unknowns, Qm and (AP)f, are determined by the two remaining equations. From Eq. (2-12),... 

The above model assumes that both components are dynamically symmetric, that they have same viscosities and densities, and that the deformations of the phase matrix is much slower than the internal rheological time [164], However, for a large class of systems, such as polymer solutions, colloidal suspension, and so on, these assumptions are not valid. To describe the phase separation in dynamically asymmetric mixtures, the model should treat the motion of each component separately ( two-fluid models [98]). Let Vi (r, t) and v2(r, t) be the velocities of components 1 and 2, respectively. Then, the basic equations for a viscoelastic model are [164—166]... 

Andrews, A. T. IV., and Sundaresan, S. Closures for filtered two-fluid model equations of gas-particle flows, Manuscript in preparation (2006). 

In the complete Eulerian description of multiphase flows, the dispersed phase may well be conceived as a second continuous phase that interpenetrates the real continuous phase, the carrier phase this approach is often referred to as two-fluid formulation. The resulting simultaneous presence of two continua is taken into account by their respective volume fractions. All other variables such as velocities need to be averaged, in some way, in proportion to their presence various techniques have been proposed to that purpose leading, however, to different formulations of the continuum equations. The method of ensemble averaging (based on a statistical average of individual realizations) is now generally accepted as most appropriate. 

In the two-fluid formulation, the motion or velocity field of each of the two continuous phases is described by its own momentum balances or NS equations (see, e.g., Rietema and Van den Akker, 1983 or Van den Akker, 1986). In both momentum balances, a phase interaction force between the two continuous phases occurs predominantly, of course with opposite sign. Two-fluid models therefore belong to the class of two-way coupling approaches. The continuum formulation of the phase interaction force should reflect the same effects as experienced by the individual particles and discussed above in the context of the Lagrangian description of dispersed two-phase flow. 

One therefore has to decide here which components of the phase interaction force (drag, virtual mass, Saffman lift, Magnus, history, stress gradients) are relevant and should be incorporated in the two sets of NS equations. The reader is referred to more specific literature, such as Oey et al. (2003), for reports on the effects of ignoring certain components of the interaction force in the two-fluid approach. The question how to model in the two-fluid formulation (lateral) dispersion of bubbles, drops, and particles in swarms is relevant... 

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. 

Then, the TDSE can be reformulated in terms of two fluid-dynamical equations ... 

Equation 9.6 and Equation 9.7 require that any interface between two fluids in equilibrium must have the constant curvature H. Appearance of two points with different curvatures results in appearance of a corresponding AP, which forces substance transfer until equalization of H. This gives rise to a... 

Having obtained two simultaneous equations for the singlet and doublet correlation functions, X and, these have to be solved. Furthermore, Kapral has pointed out that these correlations do not contain any spatial dependence at equilibrium because the direct and indirect correlations of position in an equilibrium fluid (static structures) have not been included into the psuedo-Liouville collision operators, T, [285]. Ignoring this point, Kapral then transformed the equation for the singlet density, by means of a Laplace transformation, which removes the time derivative from the equation. Using z as the Laplace transform parameter to avoid confusion with S as the solvent index, gives... 

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