Erosion

Impact Erosion

The erosion rate is defined as the mass of wall material eroded per unit area per unit time. It is calculated on wall, baffle, and contact boundary faces by accumulating the damage that each particle impact does on the face:

where:

• $A_f$ is the area of the face
• ${\dot{m}}_{\pi }$ is the mass flow rate of particles in parcel $\pi$ impacting on the face
• $e_r$ is the erosion ratio

The summation is over all parcels which strike the face in this iteration or time-step. The erosion rate therefore depends on the flow (whether and how particles impact on the wall) and the chosen method for the erosion ratio $e_r$.

Abrasive Erosion

Abrasive erosion is available for DEM only. Abrasive erosion is due to scouring, where particles scrape tangentially along walls or strike at low angles. The abrasive erosion rate is calculated as:

where:

• $\mathrm{\Delta }t$ is the fluid time-step
• $e_r$ is the erosion ratio that is calculated with the Archard correlation

Erosion Correlations

Ahlert Correlation

The Ahlert correlation [641] for the erosion ratio is:

$$e_r=K{F}_{s}f\left(\alpha \right){\left(\frac{v_{\text{rel}}}{v_{\text{ref}}}\right)}^n$$

(3308)

where:

• $K$ is a material-dependent constant
• $F_s$ is a factor that accounts for the shape of the particles
• $f\left(\alpha \right)$ is a function expressing the dependency on the particle incidence angle
• $v_{\text{rel}}$ is the magnitude of the relative velocity of the particle with respect to the wall, $\left|v_{\text{rel}}\right|,(v_{\text{rel}}=v_{\text{particle}}-v_{\text{wall}})$
• $v_{\text{ref}}$ is a constant reference velocity
• $n$ is a constant exponent

The shape coefficient $F_s$ is reported to take the value 1 for angular particles, 0.53 for semi-rounded particles and 0.2 for fully rounded particles.

The angle function $f\left(\alpha \right)$ is split into two ranges. Below the user-specified transition angle ${\alpha }_0$ it is a polynomial in $\alpha$, with the incidence angle in radians. Above the transition angle, $f\left(\alpha \right)$ follows a trigonometric relationship

$$f\left(\alpha \right)=x{\cos }^2\alpha \sin \omega \alpha +y{\sin }^2\alpha +z$$

(3309)

The constants $w$, $x$, and $y$ are user-defined, whereas $z$ is calculated internally by requiring that $f\left(\alpha \right)$ is continuous at ${\alpha }_0$.

The default coefficients for the Ahlert correlation are for liquid-borne semi-rounded sand particles eroding aluminum, as given by McLaury and others [674].

DNV Correlation

The DNV correlation [663] for the erosion ratio is:

$$e_r=Kf\left(\theta \right){\left(\frac{v_{\text{rel}}}{v_{\text{ref}}}\right)}^n$$

(3310)

in which:

• $K$ is a material-dependent constant, with a default value of 2.0E-9, for steel.
• $f\left(\theta \right)$ is a function expressing the dependency on the particle incidence angle. The default form of the function is:

$$f\theta =A\theta +B{\theta }^2+C{\theta }^3+D{\theta }^4+E{\theta }^5+F{\theta }^6+G{\theta }^7+H{\theta }^8$$

(3311)

where the values of the coefficients are:


$\begin{array}{cc}A=9.37& E=170.137\\ B=-42.295& F=-98.398\\ C=110.864& G=31.211\\ D=-175.804& H=-4.170\end{array}$

• $\theta$ is in radians and has a range of 0 to $\frac{\pi }{2}$.
• $v_{\text{rel}}$ is the magnitude of the relative velocity of the particle with respect to the wall, $\left|{\mathbf{\text{v}}}_{\text{rel}}\right|,({\mathbf{\text{v}}}_{\text{rel}}={\mathbf{\text{v}}}_{\text{particle}}-{\mathbf{\text{v}}}_{\text{wall}})$.
• $v_{\text{ref}}$ is a constant reference velocity. The default value is 1 m/s.
• $n$ a constant exponent. The default value is 2.6.

The default coefficients for the DNV correlation are for air-borne sand eroding carbon steel, and are taken from [663]. The DNV correlation does not explicitly depend on particle diameter. However, the correlation was derived from experimental data observed for a mean particle diameter of 225 µm.

Neilson-Gilchrist Correlation

The Neilson-Gilchrist [677] correlation for the erosion ratio is:

$$e_r=e_{rC}+e_{rD}$$

(3312)

in which $e_{rC}$ and $e_{rD}$ represent contributions from cutting and deformation respectively. The cutting erosion is modeled as a function of the incidence angle $\alpha$

$$e_{rC}=\{\begin{array}{cc}\frac{v_{\text{rel}}^2{\cos }^2\alpha \sin \frac{\pi \alpha }{2{\alpha }_0}}{2{\epsilon }_C}& \alpha <{\alpha }_0\\ \frac{v_{\text{rel}}^2{\cos }^2\alpha }{2{\epsilon }_C}& \alpha \ge {\alpha }_0\end{array}$$

(3313)

where:

• ${\alpha }_0$ is the user-specified transition angle.
• ${\epsilon }_C$ is the user-specified cutting coefficient.

The deformation erosion is similarly

$$e_{rD}=\frac{max{\left(u_{\text{rel}}\sin \alpha -K,0\right)}^2}{2{\epsilon }_D}$$

(3314)

with ${\epsilon }_D$ being the deformation coefficient and $K$ the cut-off velocity, below which no deformation erosion occurs.

The default coefficients for the Neilson-Gilchrist correlation are for liquid-borne sand eroding AISI 4130 steel, and are taken from Wallace and others [713].

Oka Correlation

The Oka correlation [681] and [682] for the erosion ratio is:

$$e_r=e_{90}g\left(\alpha \right){\left(\frac{v_{\text{rel}}}{v_{\text{ref}}}\right)}^{k_2}{\left(\frac{D_p}{D_{\text{ref}}}\right)}^{k_3}$$

(3315)

where:

• The angle function $g\left(\alpha \right)$ is defined as:

$$g\left(\alpha \right)={\left(\sin \alpha \right)}^{n_1}{(1+H_v(1-\sin \alpha ))}^{n_2}$$

(3316)

with $n_1$, $n_2$, and $H_v$ user-specified constants. Oka and others identify the value of $H_v$ as the Vickers hardness of the eroded material in units of GPa.

• $v_{\text{rel}}$ is the magnitude of the relative velocity of the particle with respect to the wall.
• $v_{\text{ref}}$ is the user-specified reference velocity.
• $D_{\text{ref}}$ is the user-specified reference diameter.
• $k_2$ and $k_3$ are user-specified exponents.

By inspection of Eqn. (3315) and Eqn. (3316), $e_{90}$ is revealed to be the reference erosion ratio at $v_{\text{rel}}=v_{\text{ref}}$, $D_p=D_{\text{ref}}$, and $\alpha =90°$. In Simcenter STAR-CCM+ the value of $e_{90}$ is calibrated by equating values for erosion ratio from the Oka and DNV models at 90^o impact angle, $D_p=225\mathrm{\mu }\mathrm{m}$ (DNV reference particle size), $D_{ref}=326\mathrm{\mu }\mathrm{m}$ (Oka reference particle size), and 104 m/s. This equation gives:

$$e_{90}=e_{r,DNV@90}{(D_{\mathrm{r}\mathrm{e}\mathrm{f}}/D_p)}^{k_3}$$

The purported strength of the Oka model is that the coefficients for a particular combination of eroded material and eroding material can be derived from more fundamental coefficients. These coefficients are specific to either the eroded material or the eroding material. Hence, for example, the fundamental coefficients for sand can serve as a basis for both sand-steel erosion and sand-aluminum erosion. The fundamental coefficients for the eroding material, in turn, are shown to be derivable from measurable properties of the eroding material such as its Vickers hardness.

Simcenter STAR-CCM+ bases the Oka correlation on the coefficients in Eqn. (3315) and Eqn. (3316); for more information on relating these coefficients to fundamental coefficients, see [681] and [682]. The default coefficients are for air-borne sand eroding 0.25% carbon steel, and are taken from [681] and [682], except for $e_{90}$ which is calibrated using the DNV correlation.

Archard Correlation

The Archard correlation [716] for the Abrasive Rate erosion ratio is:

$$e_r=aFs$$

(3317)

where:

• $a$ is the Abrasive Wear Coefficient, with a default value of 0.01 kg/J. The default value of 1E-2 is the high end of the recommended range of values, 1E-8 to 1E-2, mild wear to severe wear. (See Archard Properties.)
• $F$ is the normal force.
• $s$ is the sliding distance.