John B. Young

Experimental validation of condensing flow theory for a stationary cascade of steam turbine blades

The paper describes a detailed experimental and theoretical study of non-equilibrium condensing steam flow in a stationary cascade of turbine blades operating transonically. Instrumentation was installed for obtaining colour schlieren photographs of the shock wave structure, the blade surface static pressure distribution, the pitchwise variation of the mean droplet radius downstream of the cascade and the stagnation pressure loss across the cascade. Only one blade profile was tested but a comprehensive set of measurements was acquired covering a wide range of inlet steam conditions and exit Mach numbers. By careful interpretation of the data, it was possible, for the first time, to infer the thermodynamic loss due to irreversible condensation directly from experimental measurements. An elaborate comparison of the experimental data with condensing flow theory was also undertaken using a two-dimensional inviscid time-marching calculation scheme, simulating both steady and unsteady flows. Excellent agreement was obtained throughout and it can be stated with some confidence that the theory and calculation procedures used reproduce accurately all the main features of steady transonic condensing flow in stationary cascades.

Classical Nucleation Theory and Its Application to Condensing Steam Flow Calculations

Abstract The paper discusses the classical theory of the homogeneous nucleation of water droplets from supersaturated vapour and its application in predicting condensation in steam nozzles. The first part consists of a review of classical nucleation theory, focusing on the many modifications made to the original Becker-Döring theory and providing some new insights into recent developments. It is concluded that the predictive accuracy required for engineering calculations is not yet attainable with a theory derived from first principles. The areas that require most attention relate to the properties of small molecular clusters and the energy transfer processes in the non-isothermal theory. Experiments in converging-diverging nozzles provide the best means for validation at the very high nucleation rates of interest, but measurements of pressure distribution and the Sauter mean droplet radius are insufficient to provide independent checks on the separate theories of nucleation and droplet growth. Nevertheless, a judicious choice for the nucleation rate equation, in combination with a standard droplet growth model and a suitable equation of state for steam, can provide accurate predictions over a wide range of conditions. The exception is at very low pressures where there is evidence that the droplet growth rate in the nucleation zone is underestimated.

Time-marching method for the prediction of two-dimensional, unsteady flows of condensing steam

A time-accurate, two-dimensional time-marching technique is presented which can predict unsteady phenomena in condensing steam flows. Conservation equations for the mixture are solved using a variation of a well-established Euler solver, while nucleation and droplet growth calculations are performed in a Lagrangian framework by tracking particle pathlines. A special averaging technique is used to retain a polydispersion of droplet sizes, necessary for the accurate modeling of the condensation processes, without consuming excessive storage or CPU time. The basic Euler solution technique has been validated by comparison with predictions from an independent source for the unsteady flow of air in a channel. The full scheme has been used to compute nucleating flows in converging-diverging nozzles for which agreement with experiment for both steady and unsteady cases is extremely good. All the results presented are for flows in nozzles for which experimental data are available, but the scheme may also be applied to turbine cascade geometries. 26 refs.

The condensation and evaporation of liquid droplets at arbitrary Knudsen number in the presence of an inert gas

Abstract A new set of equations describing the growth and evaporation of stationary liquid droplets in a mixture of pure vapour and inert gas is presented. The equations, which model the heat and mass transfer between the droplet and its environment, are presented in a simple algebraic form and are suitable for practical calculations of droplet growth at any Knudsen number and at any concentration of inert gas. In particular, they are not restricted to the so-called quasi-steady regime of droplet growth when the droplet surface temperature has relaxed to its steady-state value. The physical model on which the theory is based is essentially that of Langmuir but some novel features are incorporated. Thus, the velocity distribution functions for vapour and inert gas molecules approaching the liquid surface are assumed to correspond to simplified Grad thirteen-moment distributions and this allows correct representation at a molecular level of the heat and mass fluxes at the outer edge of the Knudsen layer. In contrast to most simple models of condensation and evaporation, the theory predicts finite (as opposed to zero) temperature and vapour pressure jumps across the Knudsen layer in the continuum limit and shows that the former is directly proportional to the concentration of vapour present. The analysis also provides a physical interpretation for the origins of the reversed temperature gradient phenomenon in the Kundsen layer, an unusual feature predicted by more complex solutions of the Boltzmann equation itself. The transition from diffusion to kinetic control as the pure vapour limit is approached is also modelled by the theory which shows that the range of Knudsen numbers over which this occurs is of the same order as the mole fraction of inert gas present.

The condensation and evaporation of liquid droplets in a pure vapour at arbitrary Knudsen number

Abstract A new set of equations describing the growth and evaporation of small liquid droplets in a pure vapour are presented. The equations, which model both mass and heat transfer between the droplet and the vapour, are in simple closed form solution and are suitable for practical calculations at any Knudsen number. The physical model on which the theory is based is essentially that of Langmuir but some novel features are incorporated. For example, the velocity distribution function for molecules approaching the liquid surface is described by a simplified Grad, thirteen-moment distribution. The results of the analysis are in close agreement with other, more complicated and less general theories to be found in the literature. In particular, the temperature jump across the Knudsen layer in the continuum limit is accurately predicted. It is also shown that Maxwell moment methods based on the Lees, two-stream Maxwellian distribution lead to incorrect results.

The calculation of inertial particle transport in dilute gas-particle flows

Abstract The paper describes a new time-marching method for calculating two-dimensional, dilute, non-turbulent, gas-particle flows using an Eulerian formulation. The method is accurate and robust, and overcomes many of the deficiencies of other schemes reported in the literature. A particular feature is the ability to calculate the particle density field accurately even in the vicinity of discontinuities, particle-free ‘shadow’ zones and particle separations from solid surfaces. The paper discusses the ill-posedness of the Eulerian equations and describes the numerical scheme, focusing on (i) the particle boundary condition at solid surfaces, (ii) the capture of discontinuities in the particle density field, (iii) special techniques to handle shadow zones, (iv) convergence acceleration for particle flows with very small Stokes numbers and, (v) the possibility of crossing particle trajectories. Applications of the method are illustrated by calculations of particle flow over a circular cylinder and through a turbine cascade. The results agree well with the predictions of a computationally more expensive Lagrangian tracking code and the method offers the possibility of extension to include turbulent particle transport.

Particle deposition from two-dimensional turbulent gas flows

Abstract The paper is concerned with the prediction of the deposition rate of small particles from two-dimensional turbulent gas flows onto solid boundaries using a fully-Eulerian two-fluid approach. Density-weighted averaging is used to derive the ensemble-averaged particle equations which are closed with simple models for the particle turbulence correlations. A possible inconsistency in the modelling is discussed. A special method of handling the equations provides much clearer insight into the physical processes governing deposition. The solution procedure uses a formulation which can automatically capture particle-free regions and can predict surface deposition rates which may vary by several orders of magnitude. This is illustrated via calculations of deposition from turbulent channel-flow which also allow prediction of the inlet region where the particle flow is not fully-developed. Deposition from turbulent boundary layers is also considered and calculations showing the interaction between velocity slip (caused by streamline curvature), viscous drag, diffusion and turbophoresis are presented. The ability to handle complex geometries is illustrated by a calculation of deposition in a gas turbine cascade. This also provides an illustration of how inertial and diffusional deposition mechanisms work in concert and how the sum of the contributions considered separately does not represent the total deposition rate. Some preliminary calculations and experimental data on the effect of thermophoresis in non-isothermal flows are also presented.

Thermophoresis of a Spherical Particle: Reassessment, Clarification, and New Analysis

The thermophoretic force acting on a spherical particle depends on the Knudsen number and the particle-to-gas thermal conductivity ratio, and it can be estimated using various analytical and numerical methods for solving the Boltzmann equation. A substantial body of experimental data also exists. Nevertheless, the situation is not as clear as it might be and this article assesses the current predictive capabilities. First, some issues of nondimensionalization and data presentation are discussed. Then, the Grad 13-moment (G13) method of solution is examined in detail, and it is shown how the method generates a hierarchy of expressions for the thermophoretic force at low Knudsen number including all the well-known results. The non-Navier–Stokes–Fourier thermal stress and pressure-driven heat flux and their relation to the phenomenon of reversed thermophoresis are discussed. Theories of thermophoresis at arbitrary Knudsen number are then examined and it emerges that there are essentially only two theories extant. The available experimental measurements of the thermophoretic force and velocity are compared with these theories. Finally, it is shown that the G13 solution can be adapted to provide an interpolation formula for the transition regime, which gives a good approximation for practical calculations and is quantitatively very different from the commonly used prescription.

An analytical solution for the Wilson point in homogeneously nucleating flows

The calculation of conditions at the Wilson point is the key to both theoretical and numerical studies of the condensation of pure vapours by homogeneous nucleation. Nucleation and droplet growth occur in a very short period of time, during which the changes of many vapour properties due to the normal thermofluid dynamic processes are negligible compared with the change of the heat release rate. This feature is exploited in an analysis leading to an approximate solution for the maximum subcooling and other properties at the Wilson point. The analysis is general but attention is focused on the main application of interest, which is the condensation of steam in high-speed flows by homogeneous nucleation. Crucial approximations are justified over a wide range of steam pressures and the analytical results reveal the dependency of steam properties at the Wilson point on controlling parameters such as the rate of pressure decrease. A direct link is established between the steam properties at the saturation point and those at the Wilson point, which, when used in multidimensional condensation flow calculations, should remove the need for very fine meshes and excessive computing resources which are otherwise required.

Movement of deposited water on turbomachinery rotor blade surfaces

A theoretical approach for calculating the movement of liquid water following deposition onto a turbomachine rotor blade is described. Such a situation can occur during operation of an aero-engine in rain. The equation of motion of the deposited water is developed on an arbitrarily oriented plane triangular surface facet. By dividing the blade surface into a large number of facets and calculating the water trajectory over each one crossed in turn, the overall trajectory can be constructed. Apart from the centrifugal and Coriolis inertia effects, the forces acting on the water arise from the blade surface friction, and the aerodynamic shear and pressure gradient. Nondimensionalization of the equations of motion provides considerable insight and a detailed study of water flow on a flat rotating plate set at different stagger angles demonstrates the paramount importance of blade surface friction. The extreme cases of low and high blade friction are examined and it is concluded that the latter (which allows considerable mathematical generalization) is the most likely in practice. It is also shown that the aerodynamic shear force, but not the pressure force, may influence the water motion. Calculations of water movement on a low-speed compressor blade and the fan blade of a high bypass ratio aero-engine suggest that in low rotational speed situations most of the deposited water is centrifuged rapidly to the blade tip region.