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