A Unified CFD Based Approach to a Variety of Condensation Processes in a Viscous Turbulent Wet Steam Flow

A family of quasi linear mathematical models is presented and calculations made for viscous turbulent wet steam flow with a variety of condensation phenomena. These models can be applied to the analysis of equilibrium condensation, homogeneous (spontaneous) condensation, heterogeneous condensation on extraneous particles, and condensation of charged dispersed phase moving in an electrostatic field. The unified model is represented by coupled systems of gas dynamic equations for viscous turbulent two-phase flow, kinetic and electro-kinetic equations tracing out combined processes of size and charge growth, and electromagnetic field equations described an electric field with an account of self-induced in-part by a moving electrical cluster. The numerical procedure is time marching, monotone, implicit, of second order accuracy by space and time coordinates, and exhibits high resolution shock capturing ability. Viscous flow field calculations made with this procedure reveal significant influence on condensation by the shear boundary layers and wakes. Distributions of cooling rate, droplet radius and parameters of the bulk flow are predicted. Verification of the codes against known experimental data is presented.


Introduction
The theory of non-equilibrium inviscid spontaneously condensing flow has been presented in a number of reviews and monographs and applied in stationary and nonstationary airfoil, nozzle and bladed passage calculations ( [1] [2] [3] [4] [5]). Int. J. Modern Nonlinear Theory and Application Although pure condensation by homogeneous nucleation is not frequently found in nature, the concepts developed for homogeneous vapor phase nucleation form the foundation of analysis of heterogeneous condensation on extraneous pre-existing particles. Consistent methodology of solution of the related problems involves an explicit time-marching algorithm in Eulerian or Eulerian-Lagrangian frame for solving the main flow conservation equations, linked with the classical theory of nucleation and droplet growth. Little information is available in the literature on the application of droplet formation in three-dimensional flows ([5]- [11]). Correct application of viscous physical models to the two-phase non equilibrium flow features the interaction of condensation with the secondary flow, boundary layers, shock, and wake gas-dynamic systems.
In the present paper, an accurate procedure, based on Navier-Stokes equations with turbulent effect, simulated according to the Baldwin-Lomax model [12], is developed for the calculation of the condensation process in 2D flow passages.
The effect of turbulent pulsations on the kinetic equations is investigated. The problem of closing based on momentum kinetic equations (Hill's chain, [13]) is generalized for the case of turbulence and arbitrary expression of the droplet growth. There has been recent interest in modeling the process of condensation in the presence of electric field. To address this, changes in kinetic model are introduced and the model of closing is proposed.
The implicit second order accurate TVD type relaxation technique was modified to solve the Navier-Stokes system of conservation laws for the medium and condensate mass fraction. High spatial resolution is achieved by implementing the two-phase discontinuity breakdown analytical solution into the numerical technique of flux computations. The procedure was found to provide stable convergence without any artificial dissipation.

Basic Model
In the absence of velocity slip the compressible Navier-Stokes conservative equations of mass, momentum and energy for the wet steam flow take an identical form to its single-phase counterpart ( where ( ) The usual convention is assumed everywhere when a letter appearing twice in one term is regarded as being summed from 1 to 3. The dotted terms indicate partial derivatives with respect to time. The system (Equation (1) The expressions of the nuclei rate, C J , critical droplet size r * and droplet size growth rate r°, taking into account the charge, borne by the droplet, can be found in [14].
Describing a turbulent flow of a condensate we assume traditional Boussinesq's type relation, which constitutes the corresponding turbulent mass flow as proportional to the gradient of function distribution (the prime sign (' ) indi-A. S. Liberson, S. H. Hesler cates turbulent pulsation, bar above-averaging) As a result of an averaging of convective terms in Equation (3) Averaging terms, relating to nucleation rates, droplet size and charge growth needs special investigation and is omitted here. In all calculations these terms have been considered in quasi-laminar approximation.
The value of droplet charge growth, incorporated into kinetic Equation (5), is governed by elementary charge precipitation. The charge on a particle can result from two sources: field charging and diffusion charging. If a particle is in a uniform electric field, ions will travel along the electric field lines and deposit on the particle ([14] [15]). Ultimately a charge is accumulated sufficient to repel additional ions and the saturation charge is attained. Diffusion charging occurs as a result of collisions between ions and particles due to the random thermal motion of the ions. Diffusion charging is a much slower process than field charging but becomes the dominant mechanism for sub-micron particles. Generally the charging mechanism can be represented in the following unsteady form convenient to implement in the original time marching frame . The complete set of details relating to the charging phenomena may be found in [14] [15].
The final closure to determine the evolution of charge bearing by droplets can be modeled by equations of electrodynamic field, where the contribution of moving charges on electric field is taken into account (q-space density of electrical field, b, D-field and diffusion coefficients) Equations ( (5) and (6)

Equilibrium Condensation
Wet

Homogeneous Non-Equilibrium Condensation
To reduce the computation time we replace Equation (5) by a set of momentum equations. For simplicity we omit electrical terms. Multiplying (5) by r k (k = 0, 1, 2, 3) and integrating from r = r * to r =∞, we explore the usual way to reduce the problem to the more compact space of variables If droplet growth does not depend on r, the known momentum equations are derived. Usually for droplets r > 1 μm the Knudsen's expression for r° independent on r is no longer valid. To extend validity of the momentum approach to the bigger droplet size we approximate r by the linear form, r r α β° = + , converting the integral-differential Equation (7) to the following system of partial differential equations

Numerical Technique
The linked system of partial derivative equations is solved on an adaptive curvilinear mesh. Using limited extrapolations of the conservative variables, the fluxes are evaluation according to the TVD approach described, for instance, in [17] [18] [19]. The general Riemann problem for discontinuity breakdown in a two analysis grid is used. Figure 3 contains the airfoil pressure distributions on the pressure and suction sides, plotted alongside the measured wall pressure data.
The pressure rise on the suction side is associated with over-expansion at the leading edge and a rapid condensation at approximately 50% of axial chord. The first pressure rise is located at the same position as its counterpart from the superheated test. The second rise of pressure is caused by the heat release during       Submit or recommend next manuscript to SCIRP and we will provide best service for you:

Conclusions
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