Analysis of water drop erosion on turbine blades based on a nonlinear liquid–solid impact model

Abstract

Water drop erosion is regarded as one of the most serious reliability concerns in the wet steam stage of a steam turbine. One of the most challenging aspects of this problem involves the fundamental solution of the transient pressure field in the liquid drop and stress field in the metal substrate, which are coupled with each other. In this paper, we first solve the fundamental problem of high-speed liquid–solid impact, both analytically and numerically, based on a nonlinear wave model. The transient pressure distribution in liquid (include shock wave) and transient stress distribution in solid are obtained for representative water drop-1Cr13 impacts, with impact speed varying from 10 m/s to 500 m/s. The relationship between the most important parameters characterizing impact and incident speed is established. With the statistics of water drop impact on the blade, a simple fatigue model is employed in this paper to obtain the lifetime map on a blade surface under typical working conditions, in terms of both impact times and operation hours. The most dangerous water drop erosion regions and operating conditions of the steam turbine blade are deduced. These results are useful for evaluating the water drop erosion mechanisms based on the fundamental solution of liquid–solid impact.

Introduction

In the wet steam stage of a steam turbine, the small water droplets (of sub-millimeter to micron sizes) may impact the blade surface at a moderately high speed (from about 30–300 m/s); the damage caused by such repetitive impacts is known as water drop erosion, a primary concern of the mechanical reliability of turbine [1]. Water drop erosion is a complicated problem that involves close coupling between fluid and solid mechanics. From computational fluid dynamics, the two-phase flow field and relevant impact statistics of the water drops (size, speed, impact location, frequency, etc.) can be obtained using well-established techniques. The most critical outstanding challenge is to solve the fundamental problem of liquid–solid impact (Fig. 1) and obtain the characteristics of transient stress field – when such information is superimposed with the impact statistics, a comprehensive fatigue analysis can be carried out for the water drop erosion problem.

Comparing with solid–solid impact [2], [3], [4], the liquid–solid impact was less studied. An extensive review on liquid impact was given by Lesser and Field [5] which provided abundant references to the historical development of such subject. A stagnation pressure model was first employed to study liquid–solid impact [6]. The simplest liquid–solid impact problem is known as the “water hammer”, where a continuous flow of liquid with a certain speed suddenly impacts a rigid solid surface under one-dimensional (1-D) condition (Fig. 2). The “water hammer force” or “water hammer pressure” is referred to as the impact pressure on the interface between liquid and solid. Cook first derived the water hammer force [7], followed by Engel [8]. Heymann proved the hypothesis that during the impact, the shock wave is formed in the water drop and breaks away from the body of drop at a specific moment [9]; after that, the lateral jet forms and the impact pressure releases when the energy transforms to the kinetic energy of lateral jet. Such hypothesis was verified by experiment via photos of evidence of the shock wave front and the lateral jet [10], [11]. In Heymann's work [9], a simple equation of the shock speed (which varies linearly with the impact speed) was assured to solve the problem numerically. Lesser and Field [5] did not make such simplification and yet they were able to obtain an analytical solution.

Along previous analytical attempts, ref. [12] established a simple pressure model upon liquid–rigid surface interaction, and then applied that pressure to obtain stress wave propagation in an elastic half-space. Based on such model, an analysis of stress waves in several specific alloys by liquid impact loading was given [13]. Apparently, in these works, the pressure field in liquid was not simultaneously coupled with the stress wave in solid. Ref. [14] reported a theoretical model to describe the effect of capillary and viscous force on spreading of a liquid drop impinging on a rigid solid surface, and the stress field in solid was not considered. In the work by Lesser [15], the finite impedance of the solid material was shown to affect both the pressure and the angle at which jetting started [11]. A quasi-dynamic and quasi-2-D analytical model for liquid mass impacting elastic-plastic solid was established and applied that to wedge-shaped and cylindrical liquid impact [16], and the results were compared with numerical calculations based on finite element method [17]. However, the transient impact pressure and the detailed distributions of 2-D transient stress–strain fields in the solid were not solved – it will be shown in this paper that the transient peak stress and pressure are much larger than the steady state, and they are primarily responsible for water drop erosion.

With the assistance of high-speed photography, ref. [18] gave pictures of Newtonian and non-Newtonian fluids' impact on dry surfaces. The diameter of liquid was between 2.28 mm and 4.87 mm and the impact speed was less than 4 m/s. High-speed impact of microdrop on solid surface was reported [19], with liquid diameter about 100 μm and velocity around 10 m/s. Field and Lesser reported an experimental study of high-speed jets [20]. Bartolo [21] showed that when water drops impacted on a hydrophobic surface, a violent jet was shot out and the velocity of which could be up to 40 times the drop impact speed. Ref. [22] measured the impact pressure and solid surface response with impact speed up to hypersonic range (1000–4500 m/s). The length scale of such measurement is around 10 mm.

Most previous experiments focused on the pressure field in the fluid, however in the water drop erosion problem, the stress field in the solid is the most critical. Consider a spherical liquid drop impacting the surface of metal blade (Fig. 1). Since the surface material spalls off upon consecutive impact during water drop erosion, the transient peak stress causing such damage due to impact must be below the surface. For typical working conditions in the turbine, the size of water drops is typically on the order of 100 μm and the typical impact velocity is well over 100 m/s, and thus the typical impact duration is much shorter than 1 μs and it is very difficult to experimentally measure the transient stress field beneath the surface of blade material under adverse working conditions. Therefore, effective theoretical model and numerical simulation are necessary for exploring the stress field and related erosion mechanisms at such a small scale.

A fully coupled liquid–solid mechanics protocol for simulating the high-speed impact process is not yet developed to our knowledge, moreover, most previous studies focused on the pressure field in the liquid instead of the stress distribution in the solid, thus not very useful for the water drop erosion problem under investigation. For example, refs. [23], [24] analyzed the liquid–solid impact by using the Compressible-Cell-and-Marker method. Their approach did not predict the formation of shock wave and did not satisfy the water hammer force on the central axis [5]. Rosenblatt [25] simulated the procedure which a water drop impact brittle target, and argued that the maximum impact pressure should be about 2 times of the water hammer force. However, such impact pressure loading model was based on the numerical calculations of spherical water drops on rigid surface (i.e. liquid and solid deformations were decoupled) and an artificial viscosity was used [5]. Adler [26], [27] reported a three-dimensional (3-D) dynamic finite element computational model to describe the impact of a water drop (with 2 mm diameter and impact speed 305 m/s) on a solid, however the formation of shock wave was not shown. Stress wave propagation in a coated elastic specimen due to water drop impact was given in [28] based on the model established in [12], and again the solid–liquid interaction was absent. Haller [29] investigated the fluid dynamics of water drop (with 200 μm diameter and impact speed 500 m/s) impact on a rigid substrate. The fluid characteristics such as shock wave envelope, very high velocity lateral jetting, and expansion waves in the bulk of the medium were discussed, but there was no attempt to explore the stress in the solid.

In all numerical studies, since the liquid is governed by the unsteady Navier–Stokes (N–S) equations, and the solid is characterized by the unsteady elastodynamic equations, appropriate simplifications must be considered in the physical model.

Several researchers used the N–S equations of viscous fluid (the fluid model) or Euler equations of non-viscous fluid to solve the flow and pressure field in the water drop during impact [23], [24], [29]. The N–S equations include at least 4 strong nonlinear equations, and the computation cost is very expensive when solving the dynamic pressure field in liquid. The most important advantage of the fluid model is that the macroscopic deformation of the water drop can be simulated. However, after the shock wave front breaks off the edge of water drop and the lateral jet is formed, the water drop splits into small scattered pieces; under such condition, the advantage of the fluid model is limited. Since the most important liquid parameter in the water drop erosion problem is the maximum impact pressure, which appears before the shock wave front breaks off [9], [15], the most critical period is from initial contact until the moment when the shock wave front exits liquid body, which is a small fraction of a μs and the macroscopic deformation of liquid drop is negligible.

When studying the characteristics in the liquid region, many researchers used the analytical solution of 1-D linear wave equation for impact on rigid substrate (the water hammer force) [7], [8], [25] or approximate solutions of 1-D shock wave motion equation for impact on elastic substrate [30], [31], [32]. Although these steady-state solutions are explicit, the disadvantage of these solutions of linear wave equation is that they cannot describe the transient effect, which is critical during high-speed impact. For example, the peak value of impact pressure in 1-D liquid–solid impact (which is more severe than the steady-state pressure, see verification below) cannot be obtained from these analytical solutions. Moreover, the motion of shock wave in a spherical water drop is very complex and cannot be precisely described by the motion of 1-D shock wave, which will be proved in this paper. Thus, a 3-D nonlinear dynamic model is required to faithfully describe the transient impact.

Previous studies by Bowden et al. [33], [34] considered the deformation mechanisms for most classes of solids, and showed that the initial stages of impact are important for simulating the damage caused by water drop impact [35]. For water drop erosion of metal material, the ultimate objective of the research of liquid–solid impact is to obtain the stress distribution and fatigue lifetime estimation in the solid blade. Many researchers used simplified methods to analyze the solid response during liquid–solid impact. These past approaches can be approximately divided to two types.

The first type is that, after the analytical or approximate solutions of the liquid pressure is obtained, the pressure distribution is regarded as a static distribution acting on the solid. Next, the static stress distribution in the solid is solved by applying such steady-state pressure distribution as the boundary condition [36]. Several errors may appear in this approach: first, as will be shown in this paper, both the transient peak pressure in liquid and peak stress in solid are much more severe than their steady-state counterparts, and the transient stress is the primary factor for impact-related erosion [4]. Second, during a nanosecond impact, the prominent stress wave and peak stress are confined in a small region below the surface, whose location and magnitude cannot be effectively captured by the static elasticity solution.

The second type is that, after the impact pressure is obtained from the analytical or approximate solutions, the pressure distribution is decomposed to point loads [13], [28]. Next, the longitudinal wave and transverse wave arising from such point loads are analyzed. The longitudinal wave only transmits normal stress and the transverse wave only transmits shear stress in solid, and the stress at a target point can be obtained by the superposition of stress waves. Although the dynamic properties of stress field are considered in a more realistic way, the model ignores the dynamic properties of impact pressure in the liquid phase. Indeed, during an impact the acting points of impact pressure (on the interface) are moving with a speed approaching the sonic speed, and the stress at any interior material point is the superposition of space distribution and time distribution of stress waves due to pressure variation.

All previous studies described above have decoupled the analyses in liquid and solid regions, which are clearly not suitable during the high-speed impact, and a more complete 3-D dynamic elastic model coupled with dynamic liquid model is necessary. The concurrent dynamic coupling of liquid–solid properties is the main focus of this paper, in particular to obtain the maximum stress point and its motion trajectory in the solid, as well as correlating such information with the potential damage to the solid material. On the interface between liquid and solid, all transient parameters, include pressure, force, mass point position and movement, are exchanged between the two regions in situ. The liquid–solid coupling model is applied to studying water drop-solid impact in turbine engine.

As discussed above, the key to water drop erosion problem is to obtain the transient stress field inside the solid upon high-speed liquid drop impact, including the peak stress, the duration time of impact, and the position of the most dangerous material points with respect to the impact speed and drop diameter. The results should be derived from a comprehensive numerical analysis that couples fluid and solid mechanics. When such fundamental solution is combined with the statistics of water drop impact on a turbine blade, the water drop erosion problem can be analyzed and erosion rate can be predicted.

With reference to Fig. 1, when a spherical water drop impacts on a solid plane, the shock wave forms inside the water drop due to the compressibility of liquid. The water drop thus imposes a pressure distribution on the surface of the solid which varies with time and space coordinates, inducing stress waves which transmit through the solid. The most likely high pressure points are near the contact edge points or along the axis of the water drop, with possible formation and propagation of microcracks.

Liquid–solid impact is different from solid–solid impact in all dimensions. The mechanical properties of a liquid drop are distinct from a solid particle, and they are described by different governing equations. We consider the most fundamental case where a spherical water drop impacts normally on a flat solid substrate. The critical theoretical issues are the following:

• The accurate value of impact pressure in the water drop, as a function of drop size and impact velocity.

• The coupled effects of the compressibility of liquid and deformation of solid on the shock wave, impact pressure, and stress.

• The transient dynamic response in the solid and the difference between the impact stress and impact pressure values.

There are many other potential research topics of liquid–solid impact, such as the movement of shock wave and the lateral jet (after the shock wave breaks off the body of water drop), however they are less directly related to the water erosion problem, and in this paper we focus on the impact pressure in the liquid and the impact stress in the solid.

A reasonable theoretical (or physical) model of the liquid–solid impact must be established prior to numerical analysis. Due to the complexity of the liquid–solid impact, a few basic assumptions are necessary and a reliable model should meet the following criterions:

• It can describe the spherical water drop in quasi-3-D (or 2-D axisymmetric) space.

• It can consider the deformation and elasticity of a solid plane. It is important to note that thanks to the short characteristic impact duration (∼ns) on turbine blade, the deformation in the metal is elastic for high-speed impacts due to the strain rate effect [4], [5].¹ For simplicity we take the solid material to be linear elastic (as will be seen later the strain of the solid is not large during deformation) and we further assume that there is no fracture directly associated with the individual impact (erosion is generated due to fatigue of multiple impacts). This greatly simplifies the current analysis and linear elasticity theory with small deformation can be adopted for the solid, and the coupling between the Euler coordinate system in liquid and Lagrange coordinate system in solid becomes easier.

• It can describe the entire period starting from the beginning of impact, until the shock wave leaves the body. There is evidence to show that the maximum pressure appears before the shock wave leaves the body of water drop [9], [15], [35], which will be validated in this paper. After the shock wave leaves the water drop, the water drop begins to splash into small pieces and the pressure on the solid surface is reduced, but that has much less effect on the erosion of the solid material.

• It can consider the pressure in the water drop and the stress in the solid simultaneously, and can describe the coupling and difference between them. The model should also take into account the most important fluid characteristics of liquid drop.

In this paper, we establish the mathematical model and numerical algorithm that meet the above criterions. To simplify the solution and reduce the computational cost, the liquid model focuses on the main physical action (such as the formation of shock wave) and ignores the secondary factors of the high-speed impact. Both the pressure field in the water drop and the stress field in the solid are derived from the 1-D analytical model and 2-D numerical simulation. The results are used to analyze the damage to turbine under typical working conditions.

A nonlinear 3-D wave model is established for liquid–solid impact in Section 2, and the features of high-speed liquid–solid impact are discussed. Since the motion speed of shock wave is close to sonic wave speed, the most prominent physical process in the liquid is dominated by acoustics, while the secondary effect of viscosity of liquid is ignored. The governing equation of liquid becomes the wave equation that is transformed from the N–S equations.

In Section 3, we validate the nonlinear model by comparing its solution with the “minimalist” model of 1-D liquid–rigid surface impact, the water hammer force. The nonlinear wave model is then applied to the quasi-3-D (axisymmetric) impact studies in Section 4. The impact speed varies from 10 m/s to 500 m/s. The influence of the water drop diameter is also discussed. Both transient pressure distribution in the water drop (include shock wave) and transient stress field in the solid, as well as their peak values and occurrence locations, are obtained.

Finally, to demonstrate the application of the proposed nonlinear liquid–solid impact model, the results are superimposed with the statistics of water drop impact on a turbine blade and an erosion (fatigue) analysis is carried out in the last section. The general framework and nonlinear impact model developed in this paper are flexible and they may be applied to other liquid–solid impact problems encountered in practice, such as rain drop erosion.