QUASI-ONE-DIMENSIONAL MODELS

The quasi-one-dimensional model of two-phase flow in a heated capillary slot, driven by liquid vaporization from the interface, is described in Chap. 8. It takes... 

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). 

Below we consider a quasi-one-dimensional model of flow and heat transfer in a heated capillary, with hydrodynamic, thermal and capillarity effects. We estimate the influence of heat transfer on steady-state laminar flow in a heated capillary, on the shape of the interface surface and the velocity and temperature distribution along the capillary axis. 

Chapter 8 consists of the following in Sect. 8.2 the physical model of the process is described. The governing equations and conditions of the interface surface are considered in Sects. 8.3 and 8.4. In Sect. 8.5 we present the equations transformations. In Sect. 8.6 we display equations for the average parameters. The quasi-one-dimensional model is described in Sect. 8.7. Parameter distribution in characteristic zones of the heated capillary is considered in Sect. 8.8. The results of a parametrical study on flow in a heated capillary are presented in Sect. 8.9. 

Significant simplification of the governing equations may be achieved by using a quasi-one-dimensional model for the flow. Assume that (1) the ratio of meniscus depth to its radius is sufficiently small, (2) the velocity, temperature and pressure distributions in the cross-section are close to uniform, and (3) all parameters depend on the longitudinal coordinate. Differentiating Eqs. (8.32-8.35) and (8.37) we reduce the problem to the following dimensionless equations ... 

From the frame of the quasi-one-dimensional model it is possible to determine the hydrodynamic and thermal characteristics of the flow in a heated capillary, accounting for the influence of the capillary force. 

The quasi-one-dimensional model of flow in a heated micro-channel makes it possible to describe the fundamental features of two-phase capillary flow due to the heating and evaporation of the liquid. The approach developed allows one to estimate the effects of capillary, inertia, frictional and gravity forces on the shape of the interface surface, as well as the on velocity and temperature distributions. The results of the numerical solution of the system of one-dimensional mass, momentum, and energy conservation equations, and a detailed analysis of the hydrodynamic and thermal characteristic of the flow in heated capillary with evaporative interface surface have been carried out. 

The quasi-one-dimensional model described in the previous chapter is applied to the study of steady and unsteady flow regimes in heated micro-channels, as well as the boundary of steady flow domains. The effect of a number of dimensionless parameters on the velocity, temperature and pressure distributions within the domains of liquid vapor has been studied. The experimental investigation of the flow in a heated micro-channel is carried out. 

Two-phase flows in micro-channels with an evaporating meniscus, which separates the liquid and vapor regions, have been considered by Khrustalev and Faghri (1996) and Peles et al. (1998, 2000). In the latter a quasi-one-dimensional model was used to analyze the thermohydrodynamic characteristics of the flow in a heated capillary, with a distinct interface. This model takes into account the multi-stage character of the process, as well as the effect of capillary, friction and gravity forces on the flow development. The theoretical and experimental studies of the steady forced flow in a micro-channel with evaporating meniscus were carried out by Peles et al. (2001). These studies revealed the effect of a number of dimensionless parameters such as the Peclet and Jacob numbers, dimensionless heat transfer flux, etc., on the velocity, temperature and pressure distributions in the liquid and vapor regions. The structure of flow in heated micro-channels is determined by a number of factors the physical properties of fluid, its velocity, heat flux on... 

The quasi-one-dimensional model is based on the system of Eqs. (10.8-10.10) with condition (10.5-10.7) and describes the major features of the flow in the heated... 

The quasi-one-dimensional model used in the previous sections for analysis of various characteristics of fiow in a heated capillary assumes a uniform distribution of the hydrodynamical and thermal parameters in the cross-section of micro-channel. In the frame of this model, the general characteristics of the fiow with a distinct interface, such as position of the meniscus, rate evaporation and mean velocities of the liquid and its vapor, etc., can be determined for given drag and intensity of heat transfer between working fluid and wall, as well as vapor and wall. In accordance with that, the governing system of equations has to include not only the mass, momentum and energy equations but also some additional correlations that determine... 

To calculate the micro-field, the quasi-one-dimensional model by Khrustalev and Faghri (1994, 1995) is used. 

The results of calculations of the Nusselt number are presented in Fig. 10.19. Here also the data of the calculated heat transfer by the quasi-one-dimensional model by Khrustalev and Faghri (1996) is shown. The comparison of the results related to one and two-dimensional model shows that for relatively small values of wall superheat the agreement between the one and two-dimensional model is good enough (difference about 3%), whereas at large At the difference achieves 30%. 

In this section we present the system of quasi-one-dimensional equations, describing the unsteady flow in the heated capillary tube. They are valid for flows with weakly curved meniscus when the ratio of its depth to curvature radius is sufficiently small. The detailed description of a quasi-one-dimensional model of capillary flow with distinct meniscus, as well as the estimation conditions of its application for calculation of thermohydrodynamic characteristics of two-phase flow in a heated capillary are presented in the works by Peles et al. (2000,2001) and Yarin et al. (2002). In this model the set of equations including the mass, momentum and energy balances is ... 

The quasi-one-dimensional model allows analyzing the behavior of the vapor-liquid system, which undergoes small perturbations. In the frame of the linear approximation the effect of physical properties of both phases, the wall heat flux and the capillary sizes, on the flow instability is studied, and a scenario of the development of a possible processes at small and moderate Peclet number is considered. 

Amici and Thalmeier (1998) used the quasi one-dimensional model mentioned in Section 4.9.1. In their approach the presence of ferromagnetically ordered Flo layers with the magnetic moments oriented perpendicular to the tetragonal c-axis is adopted and the competition of the RKKY interaction along the c-axis with the crystalline electric field is analyzed in order to determine the transition between the commensurate antiferromagnetic structure and the incommensurate c spiral shown in Figure 39. 

P2VN(70,000)/ PS(2200), [17,18] has a higher value of E. Resolution of this problem will require development of a quasi one-dimensional model, perhaps bai. d on a random walk on a fractal lattice. Research along these lines is being pursued by Webber. [26]... 

Hall-effect thrusters exhibit a geometric configuration facilitating the use of a quasi-one-dimensional model, somewhat like the nozzles in chemical propiilsion (often, we can even adopt the one-dimensional model with a constant straight section on this subject, see Appendix A2). 

In general, the description of electrospun jets passes through quasi-one-dimensional models based on the slenderness of the system. For instance, modern models of electrospinning are also influenced by studies performed by A. M. Ganan-Calvo about steady cone-jet electrospraying systems of Newtonian liquids in the years 1997-1999. These have led to pictures by means of electro-hydrodynamic equations that have identical features as in... 

F. Pforte, A. Gerlach, A. Goldmann, R. Matzdorf, J. Braun, A. Postnikov, Wave-vector-dependent symmetry analysis of a photoemission matrix element the quasi-one-dimensional model system Cu(110)(2 x 1)0. Phys. Rev. B 63(16), 165405 (2001)... 

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