Erosion is the wearing away of mass from the surface of a body (barrier) by a flow of a fluid containing particles. Under natural conditions, the motion flow of particles is associated with the mobility of the carrier phase whether gas or liquid. Driven by inertia or other forces, the particles in a flow fall or onto the surface of the body and may cause damage. In contrast to a chemical or a thermal phenomenon (see Ablation), this is called erosion destruction.

Until 1930, studies on erosion were mainly experimental in nature, directed toward fluid and their break-up processes. More recently, the problem of wearing of steam turbine rotor blades arose, with water drops appearing in the steam circuit due to condensation. No less serious were problems involving pulverized coal in gas-turbine locomotives as well as problems in the fields of catalytic cracking, hydraulic power engineering, aviation and cosmonautics. Of particular interest was air vehicle flight through atmospheric formations (rain, snow, hail, dust) where erosion made optical and radio location devices, heat fairings and elements of control system, inoperative in a short time. Studies on the break-down of control surfaces in solid propellant rocket engineering even spawned a new term—“dust knife”—which refers to the formation of narrow zones with increased particle concentrations in carrier flows.

As knowledge of high-speed heterogeneous flows spread in scientific and technical literature, the notion of an erosion barrier (analogous to sound barrier in aviation) appeared. This concept implied that no modification of construction materials and coatings could ensure the required level of stability and availability of a unit unless the entire structure of a heterogeneous flow is deliberately changed.

Erosion destruction is the sum of many elementary processes, which complicate the interpretation of experimental data and the construction of a general physical model of this phenomenon. Several typical conditions for the interaction of particles with a streamlined body can be distinguished in a heterogeneous flow, depending on the size and velocity of particles and also on their concentration.

The impact of an isolated particle into a surface type the mechanical motion of a body encountering resistance. Unlike classical problems of hydrodynamics and ballistics, a more intricate relationship between resistance and the instantaneous velocity of a particle V_{p} exists. For a simple case of a particle colliding in a direction normal to a semi-infinite homogeneous plate, the momentum equation is expressed as:

Here, m_{p} and S_{p} are the mass of a particle and the projection of its contact surface with the barrier onto a plane perpendicular to the velocity vector V_{p} (Figure 1); H_{0} and ρ_{0} are the dynamic hardness and the density of the barrier material; C_{D} is the inertia resistance coefficient, which depends mainly on particle shape. For a sphere, ; for a cylinder with a flat face, ; for a cylinder with a conical lip, , where θ is the vertex angle of the cone.

Figure 1 shows the penetration process of a spherical particle with diameter d. Depending on the depth of penetration or the depth of crater h, the projection area of a contact spot S_{p} varies within the range of (h/d_{p}) < 0.5 as:

Denoting the density of the particle material by ρ_{p} leads to solution of Eqs. (1) and (2):

For small values of the logarithmic term is given approximately by these values and:

Equation (4) can be conveniently used for the solution of a reverse problem, i.e.: the determination of the dynamic hardness of the barrier material H_{0} from measured values of striker velocity V_{p} and crater depth h. The dynamic hardness H_{0}, in contrast to static hardness H_{B}, is not a constant parameter of the material; however, for most metals, hardness H_{0} grows only slightly with striker velocity at room temperatures, thus , where n = 0.02 – 0.04. Within the range of 10 < V_{p} < 1000 m/s, dynamic hardness H_{0} exceeds the Brunel hardness by no more than 1.5 – 2 times.

Figure 2 shows the general results of studies on collisions between iron (curve 1) and copper (curve 2) balls with a massive lead plate, and also of iron, copper and lead balls with plates of the same material (curve 3). Comparing the theoretical dependence (Eq. (3) or (4)) with the experimental data, a qualitative agreement is noted: the theoretical model for the penetration of an undistorted (solid) particle into a barrier is not effective at high levels of striker energy since it does not describe the typical “hump” and a “trough” on the curves for iron and copper balls, which strike against a lead barrier. But more important, it does not explain why all the three experimental curves merge over the range of high velocities V_{p}.

In erosion destruction, pressure develops in the zone of particle contact with the barrier, which eventually exceeds the yield point of the material. This provides a basis for the construction of a hydrodynamic analogy with a liquid jet directed into a basin filled with liquid. However, the liquid jet model poorly describes the geometrical features of compact (for instance, spherical) particles.

The limiting factors cited above have spurred the search for other approaches in constructing theoretical models of the process of particle collisions with barriers. As in the energy approach, the unknown quantity is the amount of ablated mass of the barrier material rather than the barrier depth. By reducing the determining parameters and replacing the mechanical characteristics of the material with thermodynamic ones, a unified interpretation of the experimental data has been achieved—both in single and in repeated impacts of particles on the barrier.

Figure 3 shows the experimental data for the effect of impact velocity of steel particles on the dimensional mass loss rate of lead barrier G. This important characteristic of erosion is defined as the ratio of mass loss of the barrier mer to the mass of a striking particle m_{p}: G = m_{er}/m_{p}. The dimensionless mass loss rate is directly proportional to kinetic energy over a wide range of collisions velocity V_{p}

The proportionality factor has an enthalpy dimension and is called the effective enthalpy of erosion destruction H_{er}(1). The index in (1) shows the type of the process: single or independent collisions. Although the ablated mass m_{er} can exceed by many times the mass of the striking particle m_{p}, it does not fully correspond to the entire volume of the crater. Thus in case of lead barriers, only 15% of the crater volume is formed due to mass loss. In brittle barriers, this can increase up to 50%. Most of the crater is formed due to compressive (radial) strains or the displacement of the substance beyond the original surface of the barrier (the so-called “collar”). It must be noted that the displaced-to-ablated mass ratio is not constant, especially within the range of low and moderate impact velocities V_{p}.

The relationship between independent particle collisions and repeated collisions in a steady-state flow can be described with the help of the *covering coefficient* K_{p}. This is defined as the ratio of the midship sections of all incident particles ∑_{i=n} S_{pi} to the area of the exposed surface S_{0}

If all the particles are spheres of fixed diameter d_{p} and density ρ_{p}, the covering coefficient K_{p} is proportional to a mass flux of particles :

The flow of independent particle collisions corresponds to K<<1 . But if K >>1 , then incident particles cover the entire surface of the body closely and repeatedly.

For a higher flux of particles erosion destruction loses the typical characteristics of local action; craters are absent. The denser the flow of particles and the less the size of each of them, the more complete and stronger is the analogy between erosion and ablation (see Ablation); or in the general case, between the action of single-phase gas flow and the interaction of a body with a heterogeneous dispersed medium. Nonetheless, the fundamental difference between homogeneous and heterogeneous flows cannot be forgotten. The latter has a great or zonal depth of interphase interaction with the body (Figure 4). Not only the shock wave before the barrier (1), but also the boundary layer (2), the body’s surface and the subsurface layer have no clear geometrical interpretation. The flow of high-velocity particles moves, as a rule, along trajectories which intersect the surface of the body (Figure 4), with the surface itself being highly irregular which makes the application of the notion of wall temperature T_{w} difficult. The so-called “regenerated” layer (3) in which many longitudinal transverse discontinuities are formed due to the repeated passage of shock waves, is moved to a considerable depth underneath the bottom of the craters. Not only the mechanical but also the thermophysical characteristics of this layer can be far from those of the original material (4). Under these conditions, the only correct approach is the use of the energy balance equation for relatively large elementary volumes, which embraces all three areas in Figure 4:

The supplied energy E_{p} is the sum of the kinetic energy of incident particles and the change in the heat content G_{p}C_{p}(T_{p} – T_{w}). The reflected part of energy flow E_{r} depends on the capability of particles to recoil. The dissipated energy Ed is identified with the energy of elastic waves inside the body, which transfers into heat and is absorbed by the barrier substance. Analogous with the thermal description (see Ablation) for the steady-state flow with velocity G_{er}, it can be assumed that , where C_{0} is the mean heat capacity of the barrier substance over the temperature range T_{0} to temperature T_{w} on the exposed surface. The two next terms in the balance (8) are the energy consumption for the destruction of the barrier G_{er}ΔQ_{er} and the destruction of particles G_{p}ΔQ_{p}; finally, the last term E_{b} corresponds to a blocked part of the initial energy of the flow of particles. The blocking effect arises due to the formation of a layer of reflected and ejected particles over the exposed surface.

Effective enthalpy of erosion destruction mentioned above can now be designated as the coefficient to be used before the mass loss rate when combining the second and the third terms in Eq. (8):

Experiments have shown that with a rise in impact velocity V_{p}, the energy of reflected particles decreases rapidly, and as a consequence decreases the contribution of E_{r}. Conversely, the role of blocking E_{b} can rise with increasing impact velocity V_{p} (at constant concentration of particles in the heterogeneous flow z_{p}). Finally, the analysis of experimental data given in Figure 2 shows that the contribution of the term G_{p}ΔQ_{p} is limited to a comparatively narrow range of collision velocity.

Schematic diagrams of the process of erosion development with ΔQ_{p}= 0 and E_{b} = 0 are given in Figure 5. Depicted by the solid line is the functional dependence of the dimensional mass loss rate of erosion or the intensity of erosion G on the impact velocity V_{p}.

Taking into account the capability of particle recoil from the barrier surface demands that a correcting factor η be introduced in Eq. (5)

where η = 0 for V_{p} ≤ V_{cr} and η → 1 for . The critical velocity V_{cr} is the boundary which determines the transition from elastic to plastic deformation of the barrier material. The threshold velocity value corresponds to the onset of so-called *flow* or *hydrodynamic* interaction between the colliding bodies and the surface. It is a known fact that such an interaction usually sets in when the pressure at the point of contact exceeds three or four times the yield point of the barrier material. Hence, it follows that the threshold value of velocity must exceed the critical by about two times . In view of this, note that following analytical relation for η can be written as

Figure 6 summarizes the experimental data on the effect of impact velocity of solid spherical particles (with a diameter d_{p} = 1.58 mm and made from tungsten carbide ρ_{p} = 14500 kg/m^{3}) on the erosion intensity of an aluminum barrier. As can be seen, the idealized model of the process (Figure 5) does not fully correspond to the experimental data presented in Figure 6. A measurable process of destruction begins in a flow of particles at rather low impact velocities V_{p} < V_{er}. However, the intensity of the initial destruction (up to V_{p} < V_{er} ) is so small that it is of no practical interest (G ≤ 0.001) .

A sharp bend in the dependence of G on V_{p} (Figure 6) occurs when the velocity reaches its critical value V_{p} = V_{cr} is a convenient starting point for creating the design diagram for determining this criterion. An essential change in the resistance C_{d}, and also in the pressure at the point of contact of a spherical particle, occurs when particle penetration increases to h/d_{p} ≥ 0.25. By comparing Equation (4), with the condition that , the critical value of the velocity can be approximately estimated:

Thus in the absence of blocking (E_{b} = 0), the intensity of erosion destruction is described by Eqs. (9), (10) and (11), with the only empirical constant in these equations being effective enthalpy H_{er}. As a rough estimation of the quantity H_{er}, the following relationship which relates H_{er} and the thermodynamic parameters for composite and heat shielding materials is recommended:

where φ is the fraction of reinforcing fibers and n is the number of longitudinal cross bonds in the composite material (for homogeneous materials φ = n = 1). The melting heat ΔQ_{m} and heat capacity C_{0} define the thermodynamic enthalpy for most stable (reinforcing) components of the material.

Experiments have indicated that even at constant impact velocity V_{p}, the action of a flux of particles on the barrier is not an arithmetic sum of single actions. Every preceding particle does not merely knock out a certain mass from the barrier but, by loosening them up, prepares the underlying layers for destruction. Damage studies have been carried out which provide a basis for the use of models of endurance failure, but these have still to account for the experimental results.

The result of the “loosening up” process is that there is an “incubation period” (at a given particle flux and velocity) for erosion rate to become constant. There is an analogy between this incubation period of erosion and the process of nonsteady thermal destruction (see Melting), during which almost half of the supplied energy goes into the substance heating and the other half is only absorbed in phase transformations. According to this analogy, there is a threshold value of the total kinetic energy of the particles impacted

which on being reached makes the process of erosion stable, i.e., the derivative (slope) of the dependence tends to a constant value, which does not depend on the mass of the particles which are subsequently deposited. In Eq. (12), is the mass of particles (per unit surface area) which need to be deposited before the incubation period ends:

The parameter a* is the second fundamental characteristic of erosion destruction (the first being H_{er}) of the barrier substance and is independent of impact velocity, mass, density of paticles, etc. The results of the experiments show that a* changes only slightly from one class of materials to another: 10^{5} < a* < 10^{6} J/m^{2}.

Figure 7 is a comparison of the calculated and experimental data for the dependence of the intensity of erosion of a composite material on the covering coefficient K_{p}, under the action of a flow of particles with diameter d_{p} = 0.5 × 10^{–3} m with an impact velocity of 1,800 (curve 1), 2,400 (curve 2), 3,000 (curve 3), 3,650 m/s (curve 4). Also shown by a dotted line are the results of the calculation of a threshold value of the covering coefficient for different impact velocities V_{p}, obtained with the use of Eqs. (7) and (12).

In case , the intensity of erosion depends on the number of particles that have fallen out or on the covering coefficient (Figure 7). This defines the relation between effective enthalpy Her and the quantity H_{er}(1) (Equation (1)) introduced earlier. The mass flux of a single particle when falling, is , which in the case of a spherical particle yields . Using Eqs. (5) and (12) yields

where

Accordingly, the critical velocity of the start of destruction with a single impact V_{cr} (Eq. (1)) will differ from the corresponding steady-state value V_{cr}:

This rule is indicative of the presence of the scale effect: critical velocity V_{cr} (Eq. (1)) increases in proportion to with a decrease in particle size d_{p}.

All the above results apply to the *case of the impact of particles in a direction normal to the barrier surface*. At present, there is only an elementary theoretical explanation for the process of collisions at an oblique angle. The experimental results disagree with predictions based on the law of conservation of momentum.