Laminar Two-Phase

A comparison of our first-order approximation with the exact solution is shown in Fig. 11.2. 

The rest of the chapter follows our approach to Chapters 5 and 6. First, we study laminar two-phase flow by approximate analytical means. Next, we consider the dimensionless numbers appropriate for two-phase. Finally, we correlate experimental data on turbulent two-phase in terms of these numbers. 

In the foregoing illustrative example we considered a two-phase problem with an un-steadyinterface. Here, we deal with a problem involving a spatially developing interface. 

Filmwise condensation (or evaporation) involves heat transfer to the liquid-vapor interface as well as convective flow in either or both phases. Here, we consider steady vapor condensation at the saturation temperature Ts which forms a liquid film while flowing down next to a vertical isothermal wall at Ta Ts (Fig. 113). Assuming a continuous smooth film starting at x = 0, we wish to determine the variation in film thickness S(x) and the local Nusselt number. 

The control volume enclosing the liquid film (but excluding the interface) increases in thickness with x as vapor condenses at mass flow wc. An integral balance for the conservation of mass and balance of momentum yields 

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Applicability of simple methods for obtaining G for laminar two-phase flow... 

Rovinsky, J., Brauner, N. Moalem Maron, D. (1997). Analytical solution for laminar two-phase flow in a fully eccentric core-annular configuration. International journal of multiphase flow, 23, 523-543. 

Brauner, N., Rovinsky, J., and Moalem Maron, D., Analytical Solution for Laminar-Laminar Two-Phase Flow in Circular Conduits, Special Issue, Chem. Engng. Communications (1995). 

Phase-separated PEI blends have been investigated. Combinations of PEI with polycarbonate (PC) or polycarbonate ester (PCE) copolymers have a fine, laminar two-phase morphology [36]. Combinations of PEI with polycarbonate or polyester carbonates yield a family of two-phase opaque systems that have reduced heat capability versus PEI, but show improved impact and better melt flow [37, 38]. 

Atomization. A gas or Hquid may be dispersed into another Hquid by the action of shearing or turbulent impact forces that are present in the flow field. The steady-state drop si2e represents a balance between the fluid forces tending to dismpt the drop and the forces of interfacial tension tending to oppose distortion and breakup. When the flow field is laminar the abiHty to disperse is strongly affected by the ratio of viscosities of the two phases. Dispersion, in the sense of droplet formation, does not occur when the viscosity of the dispersed phase significantly exceeds that of the dispersing medium (13). 

The basis for single-phase and some two-phase friction loss (pressure drop) for fluid flow follows the Darcy and Fanning concepts. The exact transition from laminar or dscous flow to the turbulent condition is variously identified as between a Reynolds number of 2000 and 4000. 

Essentially, except for once-through boilers, steam generation primarily involves two-phase nucleate boiling and convective boiling mechanisms (see Section 1.1). Any deposition at the heat transfer surfaces may disturb the thermal gradient resulting from the initial conduction of heat from the metal surface to the adjacent layer of slower and more laminar flow, inner-wall water and on to the higher velocity and more turbulent flow bulk water. 

The relation between c and / and X (defined by equation 5.1) is shown in Figure 5.4, where it is seen that separate curves are given according to the nature of the flow of the two phases. This relation was developed from studies on the flow in small tubes of up to 25 mm diameter with water, oils, and hydrocarbons using air at a pressure of up to 400 kN/m . For mass flowrates per unit area of U and G for the liquid and gas, respectively, Reynolds numbers Rei L d/fii ) and Rec(G d/fia) may be used as criteria for defining the flow regime values less than 1000 to 2000, however, do not necessarily imply that the fluid is in truly laminar flow. Later experimental work showed that the total pressure has an influence and data presented by Gr1H ITH(i9) may be consulted where... 

Chapter 7 deals with the practical problems. It contains the results of the general hydrodynamical and thermal characteristics corresponding to laminar flows in micro-channels of different geometry. The overall correlations for drag and heat transfer coefficients in micro-channels at single- and two-phase flows, as well as data on physical properties of selected working fluids are presented. The correlation for boiling heat transfer is also considered. 

The problems of micro-hydrodynamics were considered in different contexts (1) drag in micro-channels with a hydraulic diameter from 10 m to 10 m at laminar, transient and turbulent single-phase flows, (2) heat transfer in liquid and gas flows in small channels, and (3) two-phase flow in adiabatic and heated microchannels. The smdies performed in these directions encompass a vast class of problems related to flow of incompressible and compressible fluids in regular and irregular micro-channels under adiabatic conditions, heat transfer, as well as phase change. 

Figure 5.31 shows a comparison of the two-phase friction multiplier data with the values predicted by Eq. (5.25) with C = 5, for both phases being laminar, and with C = 0.66 given by Mishima and Hibiki s (1996) correlation. It is clear that the data correlate well using a Lockhart-Martinelli parameter, but the predictions of... 

In Table 6.7, C is the Martinelli-Chisholm constant, / is the friction factor, /f is the friction factor based on local liquid flow rate, / is the friction factor based on total flow rate as a liquid, G is the mass velocity in the micro-channel, L is the length of micro-channel, P is the pressure, AP is the pressure drop, Ptp,a is the acceleration component of two-phase pressure drop, APtp f is the frictional component of two-phase pressure drop, v is the specific volume, JCe is the thermodynamic equilibrium quality, Xvt is the Martinelli parameter based on laminar liquid-turbulent vapor flow, Xvv is the Martinelli parameter based on laminar liquid-laminar vapor flow, a is the void fraction, ji is the viscosity, p is the density, is the two-phase frictional... 

In this table the parameters are defined as follows Bo is the boiling number, d i is the hydraulic diameter, / is the friction factor, h is the local heat transfer coefficient, k is the thermal conductivity, Nu is the Nusselt number, Pr is the Prandtl number, q is the heat flux, v is the specific volume, X is the Martinelli parameter, Xvt is the Martinelli parameter for laminar liquid-turbulent vapor flow, Xw is the Martinelli parameter for laminar liquid-laminar vapor flow, Xq is thermodynamic equilibrium quality, z is the streamwise coordinate, fi is the viscosity, p is the density, <7 is the surface tension the subscripts are L for saturated fluid, LG for property difference between saturated vapor and saturated liquid, G for saturated vapor, sp for singlephase, and tp for two-phase. 

The quasi-one-dimensional model of laminar flow in a heated capillary is presented. In the frame of this model the effect of channel size, initial temperature of the working fluid, wall heat flux and gravity on two-phase capillary flow is studied. It is shown that hydrodynamical and thermal characteristics of laminar flow in a heated capillary are determined by the physical properties of the liquid and its vapor, as well as the heat flux on the wall. 

Peles el al. (2000) elaborated on a quasi-one-dimensional model of two-phase laminar flow in a heated capillary slot due to liquid evaporation from the meniscus. Subsequently this model was used for analysis of steady and unsteady flow in heated micro-channels (Peles et al. 2001 Yarin et al. 2002), as well as the study of the onset of flow instability in heated capillary flow (Hetsroni et al. 2004). 

Weislogel MM, Lichter S (1998) Capillary flow in an interior corner. 1 Eluid Mech 373 349-378 Wu PY, Little WA (1984) Measurement of the heat transfer characteristics of gas flow a fine channels heat exchangers used for microminiature refrigerators. Cryogenics 24 415 20 Xu X, Carey VP (1990) Film evaporation from a micro-grooved surface an approximate heat transfer model and its comparison with experimental data. J Thermophys 4(4) 512-520 Yarin LP, Ekelchik LA, Hetsroni G (2002) Two-phase laminar flow in a heated micro-channels. Int J Multiphase Flow 28 1589-1616... 

Peles et al. (1998) and Khrustalev and Faghri (1996) considered two-phase laminar flow in a heated micro-channel with distinct evaporating meniscus in the frame of quasi-one-dimensional and two-dimensional models. 

The developed theory of two-phase laminar flow with a distinct interface which is based on a one-dimensional approximation, takes into account the major features of the process the inertia, gravity, surface tension and friction forces and leads to the physically realistic pattern of a laminar flow in a heated micro-channel. This allows one to use the present theory to study the regimes of flow as well as optimizing a cooling system of electronic devices with high power densities. 

Yarin LP, Ekelchik LA, Hetsroni G (2002) Two-phase laminar flow in a heated micro-channels. Int J Multiphase How 28 1589-1616... 

Heat transfer in laminar single-phase, two-stream microchannels has been well described in the literature [33, 39-41]. The challenge for medium- to large-scale processes is an added complexity if more than two fluids are required and also the micromanifold region. 

A significant advance in this area was recently made by Li and coworkers [30,31], who developed a laminar flow technique, that allowed the direct contact of two liquids with better-defined mass transport compared to the Lewis cell. Laminar flow of the two phases parallel to the interface was produced through the use of flow deflectors. By forcing flow parallel to, rather than towards, the interface, it was proposed that the interface was less likely to be disrupted. Reactions were followed by sampling changes in bulk solution concentrations. 

A schematic of the apparatus developed is shown in Fig. 3. Stirrers mix and push the lighter and heavier phases in each compartment, with the maximum rotation speed governed by the need to maintain the interface steady. Flow deflectors ensure that the phases are circulated in each chamber and that flow near the interface is laminar. The interfacial plate (thickness 2 mm) is rectangular with a hole at its center. The distance from the interface to the flow deflectors is less than 6 mm. The two phases are analyzed by withdrawing small volumes via sampling holes. 

The principles of conservation of mass and momentum must be applied to each phase to determine the pressure drop and holdup in two phase systems. The differential equations used to model these principles have been solved only for laminar flows of incompressible, Newtonian fluids, with constant holdups. For this case, the momentum equations become... 

Lockhart and Martinelli divided gas-liquid flows into four cases (1) laminar gas-laminar liquid (2) turbulent gas-laminar liquid (3) laminar gas-turbulent liquid and (4) turbulent gas-turbulent liquid. They measured two-phase pressure drops and correlated the value of 0g with parameter % for each case. The authors presented a plot of acceleration effects, incompressible flow (3) no interaction at the interface and (4) the pressure drop in the gas phase equals the pressure drop in the liquid phase. 

This regime is characterized by the presence of two continuous fluid phases and an interface which can easily be described. The term separated flows is frequently employed to describe these situations in both horizontal and vertical systems. Some flow patterns in Regime I are advantageous for transferring heat between the tube wall and the fluid mixture or for carrying out two-phase reactions. The special case of laminar-laminar flow is included in this regime, and two studies seem to be of interest, Byers and King (B7) and Bentwich and Sideman (B3). 

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