Nomenclature
\(A\) = constant in Rayleigh surface wave attenuation (Pa√m) or elongation at fracture (%)
\(a\) = material parameter of strain hardening (1/Pa), or radius of the contact area (m)
\(b\) = material parameter for the residual stress (1/Pa)
\(C_{v}\) = volume concentration of water in air (-)
\(c_{l,m}\) = longitudinal wave velocity of metal (m/s)
\(c_{R}\) = Rayleigh surface wave velocity in a metal (m/s)
\(c_{t}\) = transverse wave velocity of metal (m/s)
\(c_{w}\) = speed of sound in water at the pressure \(p_{\text{wh}}\)(m/s)
\(c_{w0}\) = speed of sound in water at a pressure of 1 bar (m/s)
\(D_{h}\) = cumulative fatigue damage per hour (h-1)
\(D_{f}\) = cumulative fatigue damage at failure (-)
\(d_{d}\) = water drop diameter (m)
\(E\) = Young’s modulus (Pa)
\(f\) = fatigue cycle frequency (Hz)
\(H\) = surface hardness after peening at a certain impact velocity (Pa)
\(H_{0}\) = material hardness when stress free and without strain hardening (Pa)
\(h_{\text{tot}}\) = correction factor for the differences between fatigue test and rain impact conditions (-)
\(I_{p}\) = droplet impingement erosion incubation period (h)
\(k\) = constant for the pressure influence on the speed of sound in water (-)
\(m\) = material parameter in fatigue tests (-)
\(N_{f}\) = fatigue life (number of cycles to failure)
\(N_{i}\) = number of fatigue cycles of the incubation period or at level \(i\) (-)
\(N_{0}\) = number of specific impacts for incubation (-)
NOR = incubation resistance number (-)
\(n\) = number of tests, or exponent for the Rayleigh wave attenuation (-)
\(n_{i}\) = number of cycles due to multiple drop impact at stress level\(i\) (-)
\(p_{\text{wh}}\) = water-hammer pressure (Pa)
\(p_{wh,th}\) = threshold water-hammer pressure (Pa)
\(R\) = stress ratio in the fatigue test (-)
\(R_{d}\) = maximum erosion rate (m/s)
\(R_{e}\) = rationalized erosion rate (-)
\(R_{m}\) = tensile strength of metal (Pa)
\(R_{p0.2}\) = yield strength of metal (Pa)
\(r\) = radial coordinate (m)
\(r_{0}\) = radius of contact area when Rayleigh wave starts (m)
\(r_{\text{wh}}\) = radius of maximum contact area with the water-hammer pressure (m)
\(S_{a}\) = stress amplitude (Pa)
\(S_{D}\) = fatigue limit (Pa)
\(S_{f}\) = material parameter in fatigue tests (Pa)
\(S_{f0}\) = fatigue strength coefficient for stress free metal and without strain hardening (Pa)
\(S_{m}\) = mean stress (Pa)
\(S_{\max}\) = maximum stress in a fatigue cycle (Pa)
\(S_{\max\left(r0\right)}\) = maximum stress due to Rayleigh wave at location \(r_{0}\) (Pa)
\(s\) = standard deviation (-)
\(v_{d}\) = water droplet impact velocity on the specimen surface (m/s)
\(v_{a}\) = radial velocity of contact area boundary (m/s)
\(t\) = time (s)
\(Z\) = reduction of area (%)
\(\Phi_{v}\) = volume of impacting water drops per unit area (m/s)
\(\nu\) = Poisson constant of metal (-)
\(\rho_{m}\) = density of metal (kg/m3)
\(\rho_{w}\) = density of water (kg/m3)
\(\sigma_{R}\) = residual (compressive) stress at the surface after peening at a certain impact velocity (Pa)
\(\frac{V_{\text{water}}}{A_{e}}\) = volume of waterdrops impinged per unit exposed area (m)
\(\frac{A_{d}}{V_{d}}\) = projected area of a waterdrop divided by the volume of a waterdrop (1/m)
Introduction
Current research on droplet impingement erosion of metallic surfaces is typically related to the lower pressure stages in steam turbines where the blades suffer from erosion due to the high water content of the steam1-3. Another industrial application suffering from droplet impingement erosion is Liquified Natural Gas (LNG) transport. Large LNG drops in a partly evaporated gas are a source of droplet impact erosion of the metallic surfaces of the transfer systems4. The blades of large wind turbines also experience droplet impingement erosion, due to impact of rain. The leading edges of wind turbine blades are often protected with polyurethane coatings5,6, yet their lifetime is still relatively short. The application of flexible metallic strips to protect the leading edge might therefore become an option, thus illustrating the potential interest in controlling impingement erosion for metallic surfaces.
Droplet impingement of surfaces results, after some time, in erosion of the surface7-9,11-13. Recently, work on the modelling of drop impact-induced stresses and related wear, is presented by Slot et al.14,15, Amirzadeh et al.16,17, and Castorrini et al.18, for relatively low impact velocities (<150 m/s), for Young’s moduli that are representative for polymers and elastomers (<5 GPa) and, more importantly for fully elastic deformation during impact. The physical and metallurgical mechanisms which determine the droplet impingement erosion incubation period (\(I_{p}\)), for metallic surfaces however, are presently not fully understood1-3,5. As a result, empirical approaches are used for the assessment of droplet impingement erosion sensitive situations involving metallic surfaces1,4,9,10.
The objective of this paper is to identify and understand the physical and metallurgical mechanisms that determine the droplet impingement erosion incubation period for metallic materials. A previously developed fatigue-based model by the authors14,15 for droplet impingement erosion of polymeric surfaces is further developed for use of metallic surfaces, given the similarities in general aspects of the erosion process. The extension of the model takes into account fatigue curves of metallic materials, the effect of additional surface hardening and the effect of a residual compressive stress state at the surface due to the “water drop peening effect”. The resulting fatigue-based model for the incubation period is tested against the multi-regression fit equation of Heymann10 - the current state-of-the-art in estimating the incubation period determined from an ASTM interlaboratory test program, see Appendix A. The developed model is used for metallic surfaces in general and includes data taken from literature for stainless steel AISI 316 and aluminium 6061-T6. By quantifying the interrelation of the physical degradation mechanisms and the mechanical properties of the metals guidelines can be given for metallic surfaces with respect to droplet impingement erosion life.
Modelling