Experimental, Computational and Mathematical Analysis of Hybrid Abrasive Flow Machining Process

Abstract

The current scenario of industrialization requires need for higher productivity which is met by advanced material removal process, i.e., abrasive flow machining (AFM) in which the internal surfaces of the workpiece is machined to higher accuracy level with the help of abrasive laden media. In this paper, the conventional AFM setup has been made hybrid using electrolytic and magnetic force arrangement alongwith rotational effect in order to achieve better results in terms of material removal and surface roughness. The newly developed in-house polymer media were utilized in the process and the input parameters taken during experimentation were magnetic flux, electrolytic rod size and shape, rotational speed, polymer media, abrasive particles and extrusion pressure. It was found that the material removal and surface roughness improvement were more in electrochemo magneto rotational AFM process compared to conventional AFM process. The experimental values were in confirmation with those obtained in the optimization techniques applied, i.e., Taguchi L9 OA, Matlab fuzzy logic and GRA-PCA. In addition, the hybrid mathematical model was developed and effect of different forces occurring in the process and computational flow analysis of media have been explained.

Appendix 1: Laws Used in Magnetic Field and Time Derivative Approach

The time derivative approach in magnetic field generation is explained as under in Eqs. (26) and (27).

$$\nabla \times \vec{H}|_{t + \Delta t/2} = e_{1} \frac{{\vec{e}|_{t + \Delta t} - \vec{e}|_{t} }}{\Delta t}$$

(26)

$$\vec{e}|_{t + \Delta t/2} = \vec{e}|_{t} + \frac{\Delta t}{e}\left( {\nabla \times \vec{H}|_{t + \Delta t/2} } \right)$$

(27)

where $\vec{e}|_{t + \Delta t/2}$ is field at next time step, $\vec{e}|_{t}$ is field at previous time step, $\frac{\Delta t}{e}$ is update coefficient and $\vec{H}|_{t + \Delta t/2}$ is curl of the other field at intermediate time step. The Maxwell Eq. (27) is used to determine electromagnetic forces and the direction of the magnetic lines. The equation can be denoted mathematically as:

$$\nabla .{\text{B}} = 0$$

(28)

where B is the magnetic flux density.

$$\frac{1}{r}\frac{d}{dr}\left( {r\frac{d\varphi }{{dr}}} \right) + \frac{{d^{2} \varphi }}{{dz^{2} }} = 0$$

(29)

where $\varphi$ is the azimuthal angle, z is the height of the workpiece where the effect of magnetic field is produced. The differential Eq. (29) clearly explains the solution of azimuthal angle.

Gauss’ Law:

$\nabla .\vec{S}$ = θ

$$\nabla .\vec{S} = \frac{\partial Sx}{{\partial x}} + \frac{\partial Sy}{{\partial y}} + \frac{\partial Sz}{{\partial z}}$$

(30)

$\nabla .\vec{B}$ = 0

$$\nabla .\vec{B} = \frac{\partial Bx}{{\partial x}} + \frac{\partial By}{{\partial y}} + \frac{\partial Bz}{{\partial z}}$$

(31)

The Gauss’ law is explained in Eqs. (30) and (31)

Faraday’s Law is explained in Eq. (32).

$$\nabla \times \vec{e} = - \frac{{\partial \vec{B}}}{\partial t}$$

$$\nabla \times \vec{e} = \left( {\frac{\partial ex}{{\partial y}} - \frac{\partial ey}{{\partial z}}} \right){\text{bx}} + \left( {\frac{\partial ey}{{\partial z}} - \frac{\partial ez}{{\partial x}}} \right){\text{by}} + \left( {\frac{\partial ey}{{\partial x}} - \frac{\partial ez}{{\partial y}}} \right){\text{bz}}$$

(32)

According to Ampere’s law as shown in Eq. (33),

$$\nabla \times \vec{H} = \vec{i} + \frac{{\partial \vec{S}}}{\partial t}$$

(33)

where $\vec{i}$ is the current density. A B field induces an H field which is directly proportional to the permeability, as shown in Eq. (34).

$$\vec{B}\left( {\text{t}} \right) = \left[ {\upmu \left( {\text{t}} \right)} \right].\vec{H}\left( {\text{t}} \right)$$

(34)

where µ is the permeability index of the medium.

$$\vec{S}\left( {\text{t}} \right) = \left[ {e_{1} \left( {\text{t}} \right)} \right].\vec{e}\left( {\text{t}} \right)$$

(35)

The above Eq. (35) denotes the induction of e field due to S field which is proportional to permittivity e₁.