A generic model for prediction of kerf cross-sectional profile in multipass abrasive waterjet milling at macroscopic scale by considering the jet flow dynamics

Abstract

Multipass abrasive waterjet (AWJ) milling is preferred for achieving high material removal, surface quality, and the local form of the target feature. However, the lack of generic models for the kerf generated in multipass erosion for realizing complex features limits AWJ’s penetration into the industry. The kerf formed in multipass erosion involves a complex physical phenomenon, which is influenced by the jet characteristics and the way of material removal (MR). This work proposes a generic kerf cross-section profile (CP) prediction model in multipass milling on Ti-6Al-4 V alloy. The proposed approach integrates the jet flow dynamics (JFD) model with the analytical-based MR model, along with understanding the complex physical phenomenon involved in kerf formation towards realizing a comprehensive model. It considers the abrasive's kinetic power change with effective jet divergence, effective standoff distance, local particle impact angles made with the previous kerf, and the dynamic jet-material property, under varying jet traverse rate (V_f) and the number of passes (N). The proposed model was assessed for its generalization capability by comparing the predicted CPs against experimental ones under varying V_f and N. The results show that the evaluated abrasive kinetic power distribution obtained from the JFD model resembles the inverted kerf CPs. The comprehensive model predicted the CPs with a maximum mean absolute error of 64.5 μm and the centreline depth with a maximum error of 2.2% under the variation of V_f (1000–5000 mm/min) and N (1–9). The modelled kerf CPs conformed with the experimental ones with a 0.97 correlation coefficient.

Appendices

Appendix A

1.1 Evaluation of jet plume divergence angle outside the focusing tube

The effective jet plume diverges angle ($\varnothing$) out of the focusing tube tip that is positioned at a distance of SoD (Fig. 5) depending on the ${W}_{ef}$ can be evaluated using Eq. (A.1).

$$\varnothing = {\mathrm{tan}}^{-1}\left[\frac{{{W}_{ef}}^{k}-{d}_{f}}{2 SoD}\right]$$

(A.1)

Appendix B

2.1 Determination of jet-material property of kerf generated in a multipass abrasive waterjet milling

The jet-material property (${\varepsilon }_{k}$) of the milled kerf generated in a single-pass of the jet over the ductile material (Ti-6Al-4 V) was evaluated by considering the (i) theoretical cross-section of the kerf, and (ii) the total amount of abrasive kinetic power, ${\dot{k}}_{s}$ available over a strip S_i located at x_i = 0 as,

$${\varepsilon }_{1} =\frac{{\left.{\dot{k}}_{s}\right|}_{{x}_{i} = 0}}{{\dot{V}}_{d}} = \frac{{\left.{\dot{k}}_{s}\right|}_{{x}_{i} = 0}}{{A}_{th }{V}_{f}}$$

(B.1)

where ${A}_{th}$ is the theoretical cross-sectional area of the kerf CP that is equal to the product of maximum centerline erosion depth (${\left.{h}_{max }\right|}_{{x}_{i} = 0}$) and effective top kerf width (${W}_{ef}$) obtained at N = 1 (i.e., k = N = 1). Whereas the jet-material property varies in the multipass erosion that was modelled by considering the change in (i) jet characteristics (i.e., ${\left.{\dot{K}}_{s}^{k}\right|}_{{x}_{i} = 0}$), and (ii) the theoretical cross-sectional area of the kerf in every jet pass. Finally, considering these phenomena in the MR model to predict kerf CP in multipass erosion, the dynamic jet-material property in each jet passing over the target surface is given by Eq. (B.2).

$${\varepsilon }_{k} = \frac{{\left.{\dot{K}}_{s}^{k}\right|}_{{x}_{i} = 0}}{\left({{h}_{max}}^{k}-{{h}_{max}}^{k-1}\right) {{W}_{ef}}^{k} {V}_{f}}$$

(B.2)

where k is equal to the N^th pass of a jet.

Therefore, using Eq. (B.2), the varying jet-material property in multipass of AWJs over a workpiece is evaluated at a given set of AWJ process parameters (Table 1). Hence, the kerf CP can be predicted in each jet pass. For each traverse rate, three data points, i.e., at N = 1, 5, and 9, the h_max and the corresponding ${W}_{ef}$ were considered. Two second-order polynomials were fitted to predict the h_max as a function of N, and ${W}_{ef}$ as a function of N. The kerf CP prediction model calls these polynomials by supplying the jet pass number (k = 1, 2, …, N) towards predicting the dynamically varying ${W}_{ef}$ and h_max with the jet pass number. The data required ($N, {W}_{ef}, {h}_{max}$) for the above procedure was obtained by measuring the h_max of the kerf generated in the initial jet pass, i.e., N = 1 $\to$ (${{W}_{ef}}^{1}, {{h}_{max}}^{1}$), somewhere in the mid of N jet passes, i.e., N = 5 $\to$ (${{W}_{ef}}^{5}, {{h}_{max}}^{5}$), and at the end jet pass, i.e., N = 9 $\to$ (${{W}_{ef}}^{9},{{h}_{max}}^{9}$) using an optical microscope.

Appendix C

3.1 Evaluation of mass distribution using the distribution of particle mass flux concentration

In Eq. (23), the abrasive particle velocity, ${V}_{p}\left(x,y\right)$ at a given set of AWJ process parameters can be obtained from Eq. (21). Hence, the particle mass flux can be evaluated with the evaluated particle velocity distribution (Eq. 21), and volume fraction distribution (Eq. (14)) is expressed by Eq. (C.1).

$$\begin{array}{cc}{\dot\varphi}_p\left(x,y\right)={\rho_p\alpha}_p\left(x,y\right)V_p\left(x,y\right)&\forall\;x\;and-r_x\leq y\leq r_x\end{array}$$

(C.1)

The particle mass flux concentration can now be evaluated at all the radial positions given by Eq. (C.2).

$$\begin{array}{cc}M_p\left(x,\;y\right)=\left\{\left(\frac{{\dot\varphi}_p\left(x,y\right)}{\sum{\dot\varphi}_p\left(x,y\right)}\right)\;\right.&\forall\;x\;and-r_x\leq y\leq r_x\end{array}$$

(C.2)

Therefore, the mass flow rate distribution ${\dot{m}}_{p}\left(x,y\right)$ in the flow field outside the focusing tube is evaluated using the concentration of particle distributions (Eq. C2) that can be expressed by Eq. (C.3).

$$\begin{array}{cc}{\dot m}_p\left(x,y\right)=M_p\left(x,y\right){\dot m}_{Ep}&\forall\;x\;and-r_x\leq y\leq r_x\end{array}$$

(C.3)

where ${\dot{m}}_{Ep}$ is the constant mass flow rate of abrasive particles set in the controller during the AWJ milling experiments. Note that, according to the law of conservation of mass, the ${\dot{m}}_{Ep}$ at any jet cross-section downstream, the nozzle exit remains constant, equal to the mass flow rate at the focusing tube exit.